%------------------------------------------------------------------------------
% File     : ITP287_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_SuccPredImperative 00461_025495
% 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    : 0094_VEBT_SuccPredImperative_00461_025495 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 16830 (4723 unt;5435 typ;   0 def)
%            Number of atoms       : 28015 (10723 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 17708 (2633   ~; 517   |;1397   &)
%                                         (2913 <=>;10248  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   36 (   2 avg)
%            Number of FOOLs       : 2549 (1226 fml;1323 var)
%            Number of X terms     : 1481 (   0  [];1162 ite; 319 let)
%            Number of types       : 1521 (1520 usr)
%            Number of type conns  : 5263 (3308   >;1955   *;   0   +;   0  <<)
%            Number of predicates  :  177 ( 174 usr;   2 prp; 0-3 aty)
%            Number of functors    : 3769 (3769 usr; 635 con; 0-7 aty)
%            Number of variables   : 30865 (29910   !; 955   ?;30865   :)
% SPC      : TX0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 21:01:28.617
%------------------------------------------------------------------------------
% Could-be-implicit typings (1520)
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tff(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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tff(ty_n_t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
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tff(ty_n_t__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__List__Olist_It__Int__Oint_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Uint32__Ouint32_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Assertions__Oassn_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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tff(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Rat__Orat_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Num__Onum_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Int__Oint_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Code____Numeral__Ointeger_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Code____Numeral__Ointeger_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_I_Eo_M_Eo_J_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Uint32__Ouint32_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Uint32__Ouint32_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_Mt__Nat__Onat_J_J,type,
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tff(ty_n_t__fun_It__String__Ochar_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__String__Ochar_J,type,
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tff(ty_n_t__fun_It__Extended____Nat__Oenat_M_Eo_J,type,
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tff(ty_n_t__itself_It__Numeral____Type__Onum0_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Real__Oreal_J,type,
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tff(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
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tff(ty_n_t__List__Olist_It__Assertions__Oassn_J,type,
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tff(ty_n_t__Heap____Time____Monad__OHeap_I_Eo_J,type,
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tff(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__List__Olist_I_Eo_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(ty_n_t__Option__Ooption_It__Real__Oreal_J,type,
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tff(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_M_Eo_J,type,
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tff(ty_n_t__fun_It__Assertions__Oassn_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Assertions__Oassn_J,type,
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tff(ty_n_t__itself_It__Enum__Ofinite____3_J,type,
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tff(ty_n_t__itself_It__Enum__Ofinite____2_J,type,
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tff(ty_n_t__itself_It__Enum__Ofinite____1_J,type,
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tff(ty_n_t__Set__Oset_It__Uint32__Ouint32_J,type,
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tff(ty_n_t__Option__Ooption_It__Rat__Orat_J,type,
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tff(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
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tff(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
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tff(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
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tff(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Uint32__Ouint32_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Uint32__Ouint32_J,type,
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tff(ty_n_t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(ty_n_t__Set__Oset_It__String__Ochar_J,type,
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tff(ty_n_t__List__Olist_It__Real__Oreal_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__String__Ochar_J,type,
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tff(ty_n_t__Heap__Oarray_It__Nat__Onat_J,type,
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tff(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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tff(ty_n_t__List__Olist_It__Num__Onum_J,type,
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tff(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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tff(ty_n_t__List__Olist_It__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Real__Oreal_J,type,
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tff(ty_n_t__VEBT____Definitions__OVEBT,type,
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tff(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
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tff(ty_n_t__Set__Oset_It__Num__Onum_J,type,
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tff(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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tff(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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tff(ty_n_t__Code____Numeral__Ointeger,type,
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tff(ty_n_t__fun_It__Rat__Orat_M_Eo_J,type,
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tff(ty_n_t__fun_It__Num__Onum_M_Eo_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_M_Eo_J,type,
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tff(ty_n_t__fun_It__Int__Oint_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Int__Oint_J,type,
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tff(ty_n_t__Product____Type__Ounit,type,
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tff(ty_n_t__Extended____Nat__Oenat,type,
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tff(ty_n_t__Heap__Oarray_I_Eo_J,type,
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tff(ty_n_t__List__Olist_I_Eo_J,type,
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tff(ty_n_t__Complex__Ocomplex,type,
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tff(ty_n_t__Assertions__Oassn,type,
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tff(ty_n_t__Set__Oset_I_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_M_Eo_J,type,
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tff(ty_n_t__Uint32__Ouint32,type,
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tff(ty_n_t__String__Ochar,type,
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tff(ty_n_t__Real__Oreal,type,
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tff(ty_n_t__Rat__Orat,type,
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tff(ty_n_t__Num__Onum,type,
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tff(ty_n_t__Nat__Onat,type,
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tff(ty_n_t__Int__Oint,type,
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% Explicit typings (3915)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: nat > fun_na7530211986279412206on_nat ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: nat > fun_op8929431053563478046on_nat ).

tff(sy_c_ATP_058Lamm__aaa____,type,
    aTP_Lamm_aaa: nat > fun_ar9185468459080841407_VEBTi ).

tff(sy_c_ATP_058Lamm__aab____,type,
    aTP_Lamm_aab: nat > fun_na6954118198501441136_VEBTi ).

tff(sy_c_ATP_058Lamm__aac____,type,
    aTP_Lamm_aac: nat > fun_na3147167826068894567_VEBTi ).

tff(sy_c_ATP_058Lamm__aad____,type,
    aTP_Lamm_aad: nat > fun_na1791409581363865698_VEBTi ).

tff(sy_c_ATP_058Lamm__aae____,type,
    aTP_Lamm_aae: nat > fun_na2269434147276521438_VEBTi ).

tff(sy_c_ATP_058Lamm__aaf____,type,
    aTP_Lamm_aaf: nat > fun_na235080101556577237_VEBTi ).

tff(sy_c_ATP_058Lamm__aag____,type,
    aTP_Lamm_aag: nat > fun_na8695323739542216524_VEBTi ).

tff(sy_c_ATP_058Lamm__aah____,type,
    aTP_Lamm_aah: nat > fun_na2998378723532091971_VEBTi ).

tff(sy_c_ATP_058Lamm__aai____,type,
    aTP_Lamm_aai: nat > fun_na1238378468414741754_VEBTi ).

tff(sy_c_ATP_058Lamm__aaj____,type,
    aTP_Lamm_aaj: nat > fun_na1238378468414741754_VEBTi ).

tff(sy_c_ATP_058Lamm__aak____,type,
    aTP_Lamm_aak: nat > fun_na4431491620761096816_VEBTi ).

tff(sy_c_ATP_058Lamm__aal____,type,
    aTP_Lamm_aal: nat > fun_op246729168895827410_VEBTi ).

tff(sy_c_ATP_058Lamm__aam____,type,
    aTP_Lamm_aam: nat > fun_op3511852844878897352_VEBTi ).

tff(sy_c_ATP_058Lamm__aan____,type,
    aTP_Lamm_aan: nat > fun_op8407643897966791296_VEBTi ).

tff(sy_c_ATP_058Lamm__aao____,type,
    aTP_Lamm_aao: vEBT_VEBT > fun_na8722835333295262128_VEBTi ).

tff(sy_c_ATP_058Lamm__aap____,type,
    aTP_Lamm_aap: vEBT_VEBT > fun_na278150576929200012_VEBTi ).

tff(sy_c_ATP_058Lamm__aaq____,type,
    aTP_Lamm_aaq: vEBT_VEBT > fun_na8762902223149059366_VEBTi ).

tff(sy_c_ATP_058Lamm__aar____,type,
    aTP_Lamm_aar: nat > fun_ar6910345309256844918_VEBTi ).

tff(sy_c_ATP_058Lamm__aas____,type,
    aTP_Lamm_aas: nat > fun_na539868455704168415_VEBTi ).

tff(sy_c_ATP_058Lamm__aat____,type,
    aTP_Lamm_aat: nat > fun_na2628252815695041222_VEBTi ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: nat > fun_na8437227744162396318_VEBTi ).

tff(sy_c_ATP_058Lamm__aav____,type,
    aTP_Lamm_aav: nat > fun_na539868455704168415_VEBTi ).

tff(sy_c_ATP_058Lamm__aaw____,type,
    aTP_Lamm_aaw: nat > fun_na2628252815695041222_VEBTi ).

tff(sy_c_ATP_058Lamm__aax____,type,
    aTP_Lamm_aax: nat > fun_na8437227744162396318_VEBTi ).

tff(sy_c_ATP_058Lamm__aay____,type,
    aTP_Lamm_aay: nat > fun_na539868455704168415_VEBTi ).

tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: nat > fun_na8437227744162396318_VEBTi ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: nat > fun_op3734645406857789385on_nat ).

tff(sy_c_ATP_058Lamm__aba____,type,
    aTP_Lamm_aba: nat > fun_na539868455704168415_VEBTi ).

tff(sy_c_ATP_058Lamm__abb____,type,
    aTP_Lamm_abb: nat > fun_na8437227744162396318_VEBTi ).

tff(sy_c_ATP_058Lamm__abc____,type,
    aTP_Lamm_abc: set_nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__abd____,type,
    aTP_Lamm_abd: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__abe____,type,
    aTP_Lamm_abe: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__abf____,type,
    aTP_Lamm_abf: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__abg____,type,
    aTP_Lamm_abg: fun_nat_o > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__abh____,type,
    aTP_Lamm_abh: fun_nat_o > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__abi____,type,
    aTP_Lamm_abi: fun_nat_o > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__abj____,type,
    aTP_Lamm_abj: fun_nat_o > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__abk____,type,
    aTP_Lamm_abk: set_int > fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__abl____,type,
    aTP_Lamm_abl: set_complex > fun_se823489619503225858plex_o ).

tff(sy_c_ATP_058Lamm__abm____,type,
    aTP_Lamm_abm: set_Code_integer > fun_se5529282185252607396eger_o ).

tff(sy_c_ATP_058Lamm__abn____,type,
    aTP_Lamm_abn: set_nat > fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__abo____,type,
    aTP_Lamm_abo: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__abp____,type,
    aTP_Lamm_abp: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__abq____,type,
    aTP_Lamm_abq: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__abr____,type,
    aTP_Lamm_abr: set_Pr5949110396991348497n_assn > fun_na674329079771440817assn_o ).

tff(sy_c_ATP_058Lamm__abs____,type,
    aTP_Lamm_abs: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__abt____,type,
    aTP_Lamm_abt: set_Code_integer > fun_na1863607095336112845eger_o ).

tff(sy_c_ATP_058Lamm__abu____,type,
    aTP_Lamm_abu: set_real > fun_na5739727354687878078real_o ).

tff(sy_c_ATP_058Lamm__abv____,type,
    aTP_Lamm_abv: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__abw____,type,
    aTP_Lamm_abw: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__abx____,type,
    aTP_Lamm_abx: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__aby____,type,
    aTP_Lamm_aby: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__abz____,type,
    aTP_Lamm_abz: set_Pr5949110396991348497n_assn > fun_na674329079771440817assn_o ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: nat > fun_ar4451907433913035500on_nat ).

tff(sy_c_ATP_058Lamm__aca____,type,
    aTP_Lamm_aca: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__acb____,type,
    aTP_Lamm_acb: set_Code_integer > fun_na1863607095336112845eger_o ).

tff(sy_c_ATP_058Lamm__acc____,type,
    aTP_Lamm_acc: set_real > fun_na5739727354687878078real_o ).

tff(sy_c_ATP_058Lamm__acd____,type,
    aTP_Lamm_acd: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__ace____,type,
    aTP_Lamm_ace: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__acf____,type,
    aTP_Lamm_acf: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__acg____,type,
    aTP_Lamm_acg: fun_nat_fun_real_o ).

tff(sy_c_ATP_058Lamm__ach____,type,
    aTP_Lamm_ach: fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__aci____,type,
    aTP_Lamm_aci: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__acj____,type,
    aTP_Lamm_acj: set_int > fun_set_int_o ).

tff(sy_c_ATP_058Lamm__ack____,type,
    aTP_Lamm_ack: set_complex > fun_set_complex_o ).

tff(sy_c_ATP_058Lamm__acl____,type,
    aTP_Lamm_acl: set_Code_integer > fun_se7289222521248229150eger_o ).

tff(sy_c_ATP_058Lamm__acm____,type,
    aTP_Lamm_acm: set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__acn____,type,
    aTP_Lamm_acn: fun_Pr3238374608701176162_int_o > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__aco____,type,
    aTP_Lamm_aco: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__acp____,type,
    aTP_Lamm_acp: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__acq____,type,
    aTP_Lamm_acq: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__acr____,type,
    aTP_Lamm_acr: fun_Code_integer_o > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__acs____,type,
    aTP_Lamm_acs: fun_Pr3238374608701176162_int_o > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__act____,type,
    aTP_Lamm_act: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__acu____,type,
    aTP_Lamm_acu: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__acv____,type,
    aTP_Lamm_acv: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__acw____,type,
    aTP_Lamm_acw: fun_Code_integer_o > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__acx____,type,
    aTP_Lamm_acx: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__acy____,type,
    aTP_Lamm_acy: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__acz____,type,
    aTP_Lamm_acz: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: nat > fun_ar4689475102899419540on_nat ).

tff(sy_c_ATP_058Lamm__ada____,type,
    aTP_Lamm_ada: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__adb____,type,
    aTP_Lamm_adb: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__adc____,type,
    aTP_Lamm_adc: set_VEBT_VEBT > fun_fu1493198444879918881VEBT_o ).

tff(sy_c_ATP_058Lamm__add____,type,
    aTP_Lamm_add: set_real > fun_fu4757868175734369219real_o ).

tff(sy_c_ATP_058Lamm__ade____,type,
    aTP_Lamm_ade: set_VEBT_VEBT > fun_fu6979804496544055329VEBT_o ).

tff(sy_c_ATP_058Lamm__adf____,type,
    aTP_Lamm_adf: set_real > fun_fu8602227297639178947real_o ).

tff(sy_c_ATP_058Lamm__adg____,type,
    aTP_Lamm_adg: set_VEBT_VEBT > fun_fu1633589921238504865VEBT_o ).

tff(sy_c_ATP_058Lamm__adh____,type,
    aTP_Lamm_adh: set_real > fun_fu4472351043248340291real_o ).

tff(sy_c_ATP_058Lamm__adi____,type,
    aTP_Lamm_adi: set_VEBT_VEBT > fun_fu8552322614197228513VEBT_o ).

tff(sy_c_ATP_058Lamm__adj____,type,
    aTP_Lamm_adj: set_real > fun_fu3313596450186583811real_o ).

tff(sy_c_ATP_058Lamm__adk____,type,
    aTP_Lamm_adk: set_nat > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__adl____,type,
    aTP_Lamm_adl: set_nat > fun_fu8304507701009772099_nat_o ).

tff(sy_c_ATP_058Lamm__adm____,type,
    aTP_Lamm_adm: fun_Pr5160692531401422620nt_nat > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__adn____,type,
    aTP_Lamm_adn: fun_Pr5160692531401422620nt_nat > fun_fu7601549123499533073_int_o ).

tff(sy_c_ATP_058Lamm__ado____,type,
    aTP_Lamm_ado: fun_nat_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__adp____,type,
    aTP_Lamm_adp: fun_nat_nat > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__adq____,type,
    aTP_Lamm_adq: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__adr____,type,
    aTP_Lamm_adr: fun_int_nat > fun_fu1805223966450871909_int_o ).

tff(sy_c_ATP_058Lamm__ads____,type,
    aTP_Lamm_ads: fun_complex_nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__adt____,type,
    aTP_Lamm_adt: fun_complex_nat > fun_fu6122667721776870753plex_o ).

tff(sy_c_ATP_058Lamm__adu____,type,
    aTP_Lamm_adu: fun_Code_integer_nat > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__adv____,type,
    aTP_Lamm_adv: fun_Code_integer_nat > fun_fu2182969643484777475eger_o ).

tff(sy_c_ATP_058Lamm__adw____,type,
    aTP_Lamm_adw: fun_Pr5160692531401422620nt_nat > fun_Pr9018117161326028078_int_o ).

tff(sy_c_ATP_058Lamm__adx____,type,
    aTP_Lamm_adx: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ady____,type,
    aTP_Lamm_ady: fun_int_nat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__adz____,type,
    aTP_Lamm_adz: fun_complex_nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: nat > fun_ar5633644826702756059on_nat ).

tff(sy_c_ATP_058Lamm__aea____,type,
    aTP_Lamm_aea: fun_Code_integer_nat > fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__aeb____,type,
    aTP_Lamm_aeb: fun_Pr5160692531401422620nt_nat > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__aec____,type,
    aTP_Lamm_aec: fun_nat_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__aed____,type,
    aTP_Lamm_aed: fun_int_nat > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__aee____,type,
    aTP_Lamm_aee: fun_complex_nat > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__aef____,type,
    aTP_Lamm_aef: fun_Code_integer_nat > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__aeg____,type,
    aTP_Lamm_aeg: set_VEBT_VEBT > fun_fu6775762051927458541VEBT_o ).

tff(sy_c_ATP_058Lamm__aeh____,type,
    aTP_Lamm_aeh: set_VEBT_VEBT > fun_fu7387521734868505405VEBT_o ).

tff(sy_c_ATP_058Lamm__aei____,type,
    aTP_Lamm_aei: set_real > fun_fu1352006120924326345real_o ).

tff(sy_c_ATP_058Lamm__aej____,type,
    aTP_Lamm_aej: set_real > fun_fu9205927727435512653real_o ).

tff(sy_c_ATP_058Lamm__aek____,type,
    aTP_Lamm_aek: set_nat > fun_fu8157887890162707073_nat_o ).

tff(sy_c_ATP_058Lamm__ael____,type,
    aTP_Lamm_ael: set_nat > fun_fu1829967339744774625_nat_o ).

tff(sy_c_ATP_058Lamm__aem____,type,
    aTP_Lamm_aem: set_int > fun_fu4320861794972987337_int_o ).

tff(sy_c_ATP_058Lamm__aen____,type,
    aTP_Lamm_aen: set_int > fun_fu6214582427498676685_int_o ).

tff(sy_c_ATP_058Lamm__aeo____,type,
    aTP_Lamm_aeo: set_complex > fun_fu5921220358231560645plex_o ).

tff(sy_c_ATP_058Lamm__aep____,type,
    aTP_Lamm_aep: set_complex > fun_fu5587198656263613639plex_o ).

tff(sy_c_ATP_058Lamm__aeq____,type,
    aTP_Lamm_aeq: set_Code_integer > fun_fu2470101743130883815eger_o ).

tff(sy_c_ATP_058Lamm__aer____,type,
    aTP_Lamm_aer: set_Code_integer > fun_fu2782498379308227194eger_o ).

tff(sy_c_ATP_058Lamm__aes____,type,
    aTP_Lamm_aes: set_VEBT_VEBT > fun_fu6610162456291744429VEBT_o ).

tff(sy_c_ATP_058Lamm__aet____,type,
    aTP_Lamm_aet: set_VEBT_VEBT > fun_fu8270842325377534013VEBT_o ).

tff(sy_c_ATP_058Lamm__aeu____,type,
    aTP_Lamm_aeu: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__aev____,type,
    aTP_Lamm_aev: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__aew____,type,
    aTP_Lamm_aew: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__aex____,type,
    aTP_Lamm_aex: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__aey____,type,
    aTP_Lamm_aey: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__aez____,type,
    aTP_Lamm_aez: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: nat > fun_ar4055148783444201335on_nat ).

tff(sy_c_ATP_058Lamm__afa____,type,
    aTP_Lamm_afa: set_VEBT_VEBT > fun_fu6610162456291744429VEBT_o ).

tff(sy_c_ATP_058Lamm__afb____,type,
    aTP_Lamm_afb: set_VEBT_VEBT > fun_fu8270842325377534013VEBT_o ).

tff(sy_c_ATP_058Lamm__afc____,type,
    aTP_Lamm_afc: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__afd____,type,
    aTP_Lamm_afd: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__afe____,type,
    aTP_Lamm_afe: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__aff____,type,
    aTP_Lamm_aff: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__afg____,type,
    aTP_Lamm_afg: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__afh____,type,
    aTP_Lamm_afh: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__afi____,type,
    aTP_Lamm_afi: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__afj____,type,
    aTP_Lamm_afj: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__afk____,type,
    aTP_Lamm_afk: set_Code_integer > fun_fu8247212220001980967eger_o ).

tff(sy_c_ATP_058Lamm__afl____,type,
    aTP_Lamm_afl: set_Code_integer > fun_fu8248176645984958714eger_o ).

tff(sy_c_ATP_058Lamm__afm____,type,
    aTP_Lamm_afm: set_VEBT_VEBT > fun_fu3470462164674526041VEBT_o ).

tff(sy_c_ATP_058Lamm__afn____,type,
    aTP_Lamm_afn: set_VEBT_VEBT > fun_fu7917602684971597757VEBT_o ).

tff(sy_c_ATP_058Lamm__afo____,type,
    aTP_Lamm_afo: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__afp____,type,
    aTP_Lamm_afp: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__afq____,type,
    aTP_Lamm_afq: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__afr____,type,
    aTP_Lamm_afr: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__afs____,type,
    aTP_Lamm_afs: set_int > fun_fu3473604023002724957_int_o ).

tff(sy_c_ATP_058Lamm__aft____,type,
    aTP_Lamm_aft: set_int > fun_fu7714913403879666293_int_o ).

tff(sy_c_ATP_058Lamm__afu____,type,
    aTP_Lamm_afu: vEBT_VEBT > fun_re2335940313395306534T_VEBT ).

tff(sy_c_ATP_058Lamm__afv____,type,
    aTP_Lamm_afv: vEBT_VEBT > fun_o_5254285122974450542T_VEBT ).

tff(sy_c_ATP_058Lamm__afw____,type,
    aTP_Lamm_afw: vEBT_VEBT > fun_na2623420418515794982T_VEBT ).

tff(sy_c_ATP_058Lamm__afx____,type,
    aTP_Lamm_afx: vEBT_VEBT > fun_in2974297243791461670T_VEBT ).

tff(sy_c_ATP_058Lamm__afy____,type,
    aTP_Lamm_afy: nat > fun_VEBT_VEBT_nat ).

tff(sy_c_ATP_058Lamm__afz____,type,
    aTP_Lamm_afz: real > fun_VEBT_VEBT_real ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: nat > fun_ar3468239453911382638on_nat ).

tff(sy_c_ATP_058Lamm__aga____,type,
    aTP_Lamm_aga: assn > fun_Pr3816910165196019820n_assn ).

tff(sy_c_ATP_058Lamm__agb____,type,
    aTP_Lamm_agb: vEBT_VEBT > fun_real_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__agc____,type,
    aTP_Lamm_agc: vEBT_VEBT > fun_o_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__agd____,type,
    aTP_Lamm_agd: fun_o_fun_nat_o ).

tff(sy_c_ATP_058Lamm__age____,type,
    aTP_Lamm_age: vEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__agf____,type,
    aTP_Lamm_agf: vEBT_VEBT > fun_int_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__agg____,type,
    aTP_Lamm_agg: fun_nat_real > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__agh____,type,
    aTP_Lamm_agh: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__agi____,type,
    aTP_Lamm_agi: fun_complex_nat > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__agj____,type,
    aTP_Lamm_agj: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__agk____,type,
    aTP_Lamm_agk: fun_nat_nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__agl____,type,
    aTP_Lamm_agl: fun_nat_nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__agm____,type,
    aTP_Lamm_agm: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__agn____,type,
    aTP_Lamm_agn: fun_complex_int > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ago____,type,
    aTP_Lamm_ago: fun_nat_int > fun_nat_real ).

tff(sy_c_ATP_058Lamm__agp____,type,
    aTP_Lamm_agp: fun_int_int > fun_int_real ).

tff(sy_c_ATP_058Lamm__agq____,type,
    aTP_Lamm_agq: fun_int_int > fun_int_rat ).

tff(sy_c_ATP_058Lamm__agr____,type,
    aTP_Lamm_agr: fun_int_int > fun_in8913651378585746847l_num1 ).

tff(sy_c_ATP_058Lamm__ags____,type,
    aTP_Lamm_ags: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__agt____,type,
    aTP_Lamm_agt: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__agu____,type,
    aTP_Lamm_agu: fun_complex_real > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__agv____,type,
    aTP_Lamm_agv: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__agw____,type,
    aTP_Lamm_agw: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__agx____,type,
    aTP_Lamm_agx: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__agy____,type,
    aTP_Lamm_agy: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__agz____,type,
    aTP_Lamm_agz: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: nat > fun_ar7932181670253460709on_nat ).

tff(sy_c_ATP_058Lamm__aha____,type,
    aTP_Lamm_aha: code_integer > fun_fu555566762037042313r_real ).

tff(sy_c_ATP_058Lamm__ahb____,type,
    aTP_Lamm_ahb: vEBT_VEBT > fun_fu2188483506383969923BT_rat ).

tff(sy_c_ATP_058Lamm__ahc____,type,
    aTP_Lamm_ahc: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__ahd____,type,
    aTP_Lamm_ahd: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ahe____,type,
    aTP_Lamm_ahe: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__ahf____,type,
    aTP_Lamm_ahf: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__ahg____,type,
    aTP_Lamm_ahg: vEBT_VEBT > fun_fu8198533418994508931T_real ).

tff(sy_c_ATP_058Lamm__ahh____,type,
    aTP_Lamm_ahh: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__ahi____,type,
    aTP_Lamm_ahi: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__ahj____,type,
    aTP_Lamm_ahj: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__ahk____,type,
    aTP_Lamm_ahk: code_integer > fun_fu555566762037042313r_real ).

tff(sy_c_ATP_058Lamm__ahl____,type,
    aTP_Lamm_ahl: vEBT_VEBT > fun_fu2188483506383969923BT_rat ).

tff(sy_c_ATP_058Lamm__ahm____,type,
    aTP_Lamm_ahm: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__ahn____,type,
    aTP_Lamm_ahn: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__aho____,type,
    aTP_Lamm_aho: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__ahp____,type,
    aTP_Lamm_ahp: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__ahq____,type,
    aTP_Lamm_ahq: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ahr____,type,
    aTP_Lamm_ahr: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ahs____,type,
    aTP_Lamm_ahs: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__aht____,type,
    aTP_Lamm_aht: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ahu____,type,
    aTP_Lamm_ahu: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__ahv____,type,
    aTP_Lamm_ahv: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ahw____,type,
    aTP_Lamm_ahw: fun_nat_fun_nat_nat > fun_se1179353343636554400at_nat ).

tff(sy_c_ATP_058Lamm__ahx____,type,
    aTP_Lamm_ahx: fun_co7673397065115134084omplex > fun_se7512233253626865466omplex ).

tff(sy_c_ATP_058Lamm__ahy____,type,
    aTP_Lamm_ahy: fun_co7673397065115134084omplex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__ahz____,type,
    aTP_Lamm_ahz: fun_co7673397065115134084omplex > fun_se7512233253626865466omplex ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_op5783775701855621896on_nat ).

tff(sy_c_ATP_058Lamm__aia____,type,
    aTP_Lamm_aia: fun_nat_fun_nat_real > fun_se1654316830156123260t_real ).

tff(sy_c_ATP_058Lamm__aib____,type,
    aTP_Lamm_aib: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__aic____,type,
    aTP_Lamm_aic: fun_nat_fun_nat_real > fun_se1654316830156123260t_real ).

tff(sy_c_ATP_058Lamm__aid____,type,
    aTP_Lamm_aid: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__aie____,type,
    aTP_Lamm_aie: fun_int_fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__aif____,type,
    aTP_Lamm_aif: fun_int_fun_int_int > fun_se4827636401813558708nt_int ).

tff(sy_c_ATP_058Lamm__aig____,type,
    aTP_Lamm_aig: fun_complex_complex > fun_complex_real ).

tff(sy_c_ATP_058Lamm__aih____,type,
    aTP_Lamm_aih: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__aii____,type,
    aTP_Lamm_aii: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__aij____,type,
    aTP_Lamm_aij: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__aik____,type,
    aTP_Lamm_aik: fun_nat_nat > fun_fu2454420574033104442at_nat ).

tff(sy_c_ATP_058Lamm__ail____,type,
    aTP_Lamm_ail: fun_complex_complex > fun_fu8998703528969179354omplex ).

tff(sy_c_ATP_058Lamm__aim____,type,
    aTP_Lamm_aim: fun_complex_complex > fun_fu8084223252616112272omplex ).

tff(sy_c_ATP_058Lamm__ain____,type,
    aTP_Lamm_ain: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__aio____,type,
    aTP_Lamm_aio: fun_nat_real > fun_fu3306995184869731826t_real ).

tff(sy_c_ATP_058Lamm__aip____,type,
    aTP_Lamm_aip: fun_int_int > fun_fu923337204662071120nt_int ).

tff(sy_c_ATP_058Lamm__aiq____,type,
    aTP_Lamm_aiq: fun_int_int > fun_fu3739269683040691718nt_int ).

tff(sy_c_ATP_058Lamm__air____,type,
    aTP_Lamm_air: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ais____,type,
    aTP_Lamm_ais: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__ait____,type,
    aTP_Lamm_ait: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__aiu____,type,
    aTP_Lamm_aiu: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__aiv____,type,
    aTP_Lamm_aiv: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__aiw____,type,
    aTP_Lamm_aiw: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__aix____,type,
    aTP_Lamm_aix: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__aiy____,type,
    aTP_Lamm_aiy: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__aiz____,type,
    aTP_Lamm_aiz: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_o_4049874799390722306on_nat ).

tff(sy_c_ATP_058Lamm__aja____,type,
    aTP_Lamm_aja: fun_complex_complex > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ajb____,type,
    aTP_Lamm_ajb: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__ajc____,type,
    aTP_Lamm_ajc: set_nat > fun_fu2435797646934184285_nat_o ).

tff(sy_c_ATP_058Lamm__ajd____,type,
    aTP_Lamm_ajd: set_nat > fun_fu2828149587716756511BT_nat ).

tff(sy_c_ATP_058Lamm__aje____,type,
    aTP_Lamm_aje: fun_VE597728521808591204at_nat > fun_na6208021027755453088BT_nat ).

tff(sy_c_ATP_058Lamm__ajf____,type,
    aTP_Lamm_ajf: set_VEBT_VEBT > fun_fu4061198407372677825at_nat ).

tff(sy_c_ATP_058Lamm__ajg____,type,
    aTP_Lamm_ajg: set_nat > fun_fu5289434066392675523_nat_o ).

tff(sy_c_ATP_058Lamm__ajh____,type,
    aTP_Lamm_ajh: set_nat > fun_fu8695033588447641837al_nat ).

tff(sy_c_ATP_058Lamm__aji____,type,
    aTP_Lamm_aji: fun_real_fun_nat_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__ajj____,type,
    aTP_Lamm_ajj: set_real > fun_fu5553606195750802321at_nat ).

tff(sy_c_ATP_058Lamm__ajk____,type,
    aTP_Lamm_ajk: set_nat > fun_fu927436805355353155_nat_o ).

tff(sy_c_ATP_058Lamm__ajl____,type,
    aTP_Lamm_ajl: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__ajm____,type,
    aTP_Lamm_ajm: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__ajn____,type,
    aTP_Lamm_ajn: set_int > fun_fu7493154764834273859_int_o ).

tff(sy_c_ATP_058Lamm__ajo____,type,
    aTP_Lamm_ajo: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__ajp____,type,
    aTP_Lamm_ajp: set_nat > fun_fu1580565752866428867_nat_o ).

tff(sy_c_ATP_058Lamm__ajq____,type,
    aTP_Lamm_ajq: set_nat > fun_fu8020931516743756395ex_nat ).

tff(sy_c_ATP_058Lamm__ajr____,type,
    aTP_Lamm_ajr: fun_co636942507509095880at_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__ajs____,type,
    aTP_Lamm_ajs: set_complex > fun_fu3074144567237359299plex_o ).

tff(sy_c_ATP_058Lamm__ajt____,type,
    aTP_Lamm_ajt: set_complex > fun_fu3351286688164852109at_nat ).

tff(sy_c_ATP_058Lamm__aju____,type,
    aTP_Lamm_aju: set_nat > fun_fu8096999557949053699_nat_o ).

tff(sy_c_ATP_058Lamm__ajv____,type,
    aTP_Lamm_ajv: set_nat > fun_fu4176442768802536380er_nat ).

tff(sy_c_ATP_058Lamm__ajw____,type,
    aTP_Lamm_ajw: fun_Co3312552843764780407at_nat > fun_na5492297256294505719er_nat ).

tff(sy_c_ATP_058Lamm__ajx____,type,
    aTP_Lamm_ajx: set_Code_integer > fun_fu6309186030181755779eger_o ).

tff(sy_c_ATP_058Lamm__ajy____,type,
    aTP_Lamm_ajy: set_Code_integer > fun_fu1095011443884336175at_nat ).

tff(sy_c_ATP_058Lamm__ajz____,type,
    aTP_Lamm_ajz: set_complex > fun_fu720419882121075617plex_o ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: nat > fun_ar2428238364875509412on_nat ).

tff(sy_c_ATP_058Lamm__aka____,type,
    aTP_Lamm_aka: set_complex > fun_fu8871020438547428127omplex ).

tff(sy_c_ATP_058Lamm__akb____,type,
    aTP_Lamm_akb: fun_VE2006399705683203104omplex > fun_co3541073130412562976omplex ).

tff(sy_c_ATP_058Lamm__akc____,type,
    aTP_Lamm_akc: set_VEBT_VEBT > fun_fu8266604298679398363omplex ).

tff(sy_c_ATP_058Lamm__akd____,type,
    aTP_Lamm_akd: set_complex > fun_fu3043630359712370243plex_o ).

tff(sy_c_ATP_058Lamm__ake____,type,
    aTP_Lamm_ake: set_complex > fun_fu7433039812794084273omplex ).

tff(sy_c_ATP_058Lamm__akf____,type,
    aTP_Lamm_akf: fun_re7208826350674946562omplex > fun_co286204877037288194omplex ).

tff(sy_c_ATP_058Lamm__akg____,type,
    aTP_Lamm_akg: set_real > fun_fu1853463631764337203omplex ).

tff(sy_c_ATP_058Lamm__akh____,type,
    aTP_Lamm_akh: set_complex > fun_fu5273811730930669507plex_o ).

tff(sy_c_ATP_058Lamm__aki____,type,
    aTP_Lamm_aki: set_complex > fun_fu6938252901338612493omplex ).

tff(sy_c_ATP_058Lamm__akj____,type,
    aTP_Lamm_akj: fun_na1591161823130732966omplex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__akk____,type,
    aTP_Lamm_akk: set_nat > fun_fu3780232916559739075_nat_o ).

tff(sy_c_ATP_058Lamm__akl____,type,
    aTP_Lamm_akl: set_nat > fun_fu3656911019502402539omplex ).

tff(sy_c_ATP_058Lamm__akm____,type,
    aTP_Lamm_akm: set_complex > fun_fu4149201653251398851plex_o ).

tff(sy_c_ATP_058Lamm__akn____,type,
    aTP_Lamm_akn: set_complex > fun_fu7724386853639975089omplex ).

tff(sy_c_ATP_058Lamm__ako____,type,
    aTP_Lamm_ako: fun_in1307110748760623106omplex > fun_co7673900812888889602omplex ).

tff(sy_c_ATP_058Lamm__akp____,type,
    aTP_Lamm_akp: set_int > fun_fu2620374121714387907_int_o ).

tff(sy_c_ATP_058Lamm__akq____,type,
    aTP_Lamm_akq: set_int > fun_fu260611630652282675omplex ).

tff(sy_c_ATP_058Lamm__akr____,type,
    aTP_Lamm_akr: set_complex > fun_fu7012200369710426499plex_o ).

tff(sy_c_ATP_058Lamm__aks____,type,
    aTP_Lamm_aks: set_complex > fun_fu403589845504736640omplex ).

tff(sy_c_ATP_058Lamm__akt____,type,
    aTP_Lamm_akt: fun_Co8905303033605938227omplex > fun_co1305099417915878579omplex ).

tff(sy_c_ATP_058Lamm__aku____,type,
    aTP_Lamm_aku: set_Code_integer > fun_fu9120999634463783171eger_o ).

tff(sy_c_ATP_058Lamm__akv____,type,
    aTP_Lamm_akv: set_Code_integer > fun_fu243871637272208593omplex ).

tff(sy_c_ATP_058Lamm__akw____,type,
    aTP_Lamm_akw: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__akx____,type,
    aTP_Lamm_akx: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__aky____,type,
    aTP_Lamm_aky: fun_VE2989161857680785728t_real > fun_na2714374887907562620T_real ).

tff(sy_c_ATP_058Lamm__akz____,type,
    aTP_Lamm_akz: set_VEBT_VEBT > fun_fu830247258886439116t_real ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: nat > fun_ar5961856363738179336on_nat ).

tff(sy_c_ATP_058Lamm__ala____,type,
    aTP_Lamm_ala: fun_int_fun_nat_real > fun_nat_fun_int_real ).

tff(sy_c_ATP_058Lamm__alb____,type,
    aTP_Lamm_alb: set_int > fun_fu7141299348810748056t_real ).

tff(sy_c_ATP_058Lamm__alc____,type,
    aTP_Lamm_alc: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ald____,type,
    aTP_Lamm_ald: set_real > fun_fu5104398059618711576t_real ).

tff(sy_c_ATP_058Lamm__ale____,type,
    aTP_Lamm_ale: fun_na1142814060743658952omplex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__alf____,type,
    aTP_Lamm_alf: set_nat > fun_fu7808106308057563584omplex ).

tff(sy_c_ATP_058Lamm__alg____,type,
    aTP_Lamm_alg: fun_VE6440932821983644738omplex > fun_na5998776666739148158omplex ).

tff(sy_c_ATP_058Lamm__alh____,type,
    aTP_Lamm_alh: set_VEBT_VEBT > fun_fu6112804207117106256omplex ).

tff(sy_c_ATP_058Lamm__ali____,type,
    aTP_Lamm_ali: fun_in8056368165505483812omplex > fun_na5558608354090810916omplex ).

tff(sy_c_ATP_058Lamm__alj____,type,
    aTP_Lamm_alj: set_int > fun_fu6465309499045537820omplex ).

tff(sy_c_ATP_058Lamm__alk____,type,
    aTP_Lamm_alk: fun_re2084383175013312036omplex > fun_na1957955181367619364omplex ).

tff(sy_c_ATP_058Lamm__all____,type,
    aTP_Lamm_all: set_real > fun_fu847012557204443036omplex ).

tff(sy_c_ATP_058Lamm__alm____,type,
    aTP_Lamm_alm: set_nat > fun_fu7979797992566127108at_nat ).

tff(sy_c_ATP_058Lamm__aln____,type,
    aTP_Lamm_aln: fun_co3258106519541737638omplex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__alo____,type,
    aTP_Lamm_alo: set_complex > fun_fu689363623238371998omplex ).

tff(sy_c_ATP_058Lamm__alp____,type,
    aTP_Lamm_alp: set_nat > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__alq____,type,
    aTP_Lamm_alq: set_VEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__alr____,type,
    aTP_Lamm_alr: fun_VEBT_VEBT_real > fun_fu4911191927098811621T_real ).

tff(sy_c_ATP_058Lamm__als____,type,
    aTP_Lamm_als: set_real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__alt____,type,
    aTP_Lamm_alt: fun_real_real > fun_fu4478067837645853357l_real ).

tff(sy_c_ATP_058Lamm__alu____,type,
    aTP_Lamm_alu: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__alv____,type,
    aTP_Lamm_alv: fun_int_real > fun_fu7030473000426307245t_real ).

tff(sy_c_ATP_058Lamm__alw____,type,
    aTP_Lamm_alw: set_complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__alx____,type,
    aTP_Lamm_alx: fun_complex_real > fun_fu3043978019632300333x_real ).

tff(sy_c_ATP_058Lamm__aly____,type,
    aTP_Lamm_aly: set_Code_integer > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__alz____,type,
    aTP_Lamm_alz: fun_Co3960896333048504454r_real > fun_fu8352104517646067181r_real ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: nat > fun_ar2366578190170504826on_nat ).

tff(sy_c_ATP_058Lamm__ama____,type,
    aTP_Lamm_ama: fun_VEBT_VEBT_rat > fun_fu14152356423011729BT_rat ).

tff(sy_c_ATP_058Lamm__amb____,type,
    aTP_Lamm_amb: fun_real_rat > fun_fu5551332503302594393al_rat ).

tff(sy_c_ATP_058Lamm__amc____,type,
    aTP_Lamm_amc: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__amd____,type,
    aTP_Lamm_amd: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__ame____,type,
    aTP_Lamm_ame: fun_int_rat > fun_fu3922030663826099033nt_rat ).

tff(sy_c_ATP_058Lamm__amf____,type,
    aTP_Lamm_amf: fun_complex_rat > fun_fu4396998534562958809ex_rat ).

tff(sy_c_ATP_058Lamm__amg____,type,
    aTP_Lamm_amg: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__amh____,type,
    aTP_Lamm_amh: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ami____,type,
    aTP_Lamm_ami: fun_VE2989161857680785728t_real > fun_VEBT_VEBT_real ).

tff(sy_c_ATP_058Lamm__amj____,type,
    aTP_Lamm_amj: fun_int_fun_nat_real > fun_int_real ).

tff(sy_c_ATP_058Lamm__amk____,type,
    aTP_Lamm_amk: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__aml____,type,
    aTP_Lamm_aml: fun_na1142814060743658952omplex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__amm____,type,
    aTP_Lamm_amm: fun_VE6440932821983644738omplex > fun_VE5569070535936029429omplex ).

tff(sy_c_ATP_058Lamm__amn____,type,
    aTP_Lamm_amn: fun_in8056368165505483812omplex > fun_int_complex ).

tff(sy_c_ATP_058Lamm__amo____,type,
    aTP_Lamm_amo: fun_re2084383175013312036omplex > fun_real_complex ).

tff(sy_c_ATP_058Lamm__amp____,type,
    aTP_Lamm_amp: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__amq____,type,
    aTP_Lamm_amq: fun_co3258106519541737638omplex > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__amr____,type,
    aTP_Lamm_amr: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ams____,type,
    aTP_Lamm_ams: fun_int_real > fun_int_o ).

tff(sy_c_ATP_058Lamm__amt____,type,
    aTP_Lamm_amt: fun_complex_real > fun_complex_o ).

tff(sy_c_ATP_058Lamm__amu____,type,
    aTP_Lamm_amu: fun_Co3960896333048504454r_real > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__amv____,type,
    aTP_Lamm_amv: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__amw____,type,
    aTP_Lamm_amw: fun_complex_rat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__amx____,type,
    aTP_Lamm_amx: fun_Code_integer_rat > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__amy____,type,
    aTP_Lamm_amy: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__amz____,type,
    aTP_Lamm_amz: fun_complex_nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: nat > fun_ar8351020206311831834on_nat ).

tff(sy_c_ATP_058Lamm__ana____,type,
    aTP_Lamm_ana: fun_Code_integer_nat > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__anb____,type,
    aTP_Lamm_anb: fun_complex_int > fun_complex_o ).

tff(sy_c_ATP_058Lamm__anc____,type,
    aTP_Lamm_anc: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__and____,type,
    aTP_Lamm_and: code_integer > fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__ane____,type,
    aTP_Lamm_ane: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__anf____,type,
    aTP_Lamm_anf: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ang____,type,
    aTP_Lamm_ang: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__anh____,type,
    aTP_Lamm_anh: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__ani____,type,
    aTP_Lamm_ani: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__anj____,type,
    aTP_Lamm_anj: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ank____,type,
    aTP_Lamm_ank: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__anl____,type,
    aTP_Lamm_anl: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__anm____,type,
    aTP_Lamm_anm: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ann____,type,
    aTP_Lamm_ann: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ano____,type,
    aTP_Lamm_ano: vEBT_VEBT > fun_fu4141963935195952137T_real ).

tff(sy_c_ATP_058Lamm__anp____,type,
    aTP_Lamm_anp: complex > fun_fu709861064203714431x_real ).

tff(sy_c_ATP_058Lamm__anq____,type,
    aTP_Lamm_anq: code_integer > fun_fu8135725022974991116r_real ).

tff(sy_c_ATP_058Lamm__anr____,type,
    aTP_Lamm_anr: vEBT_VEBT > fun_fu4463953228700081077BT_rat ).

tff(sy_c_ATP_058Lamm__ans____,type,
    aTP_Lamm_ans: complex > fun_fu6638204120336560003ex_rat ).

tff(sy_c_ATP_058Lamm__ant____,type,
    aTP_Lamm_ant: code_integer > fun_fu8767785434844327952er_rat ).

tff(sy_c_ATP_058Lamm__anu____,type,
    aTP_Lamm_anu: vEBT_VEBT > fun_fu5586652545743564717BT_nat ).

tff(sy_c_ATP_058Lamm__anv____,type,
    aTP_Lamm_anv: complex > fun_fu2513798379308225899ex_nat ).

tff(sy_c_ATP_058Lamm__anw____,type,
    aTP_Lamm_anw: code_integer > fun_fu2498091903632033272er_nat ).

tff(sy_c_ATP_058Lamm__anx____,type,
    aTP_Lamm_anx: vEBT_VEBT > fun_fu7059288016784409353BT_int ).

tff(sy_c_ATP_058Lamm__any____,type,
    aTP_Lamm_any: fun_VEBT_VEBT_int > fun_fu5786399312697532803BT_int ).

tff(sy_c_ATP_058Lamm__anz____,type,
    aTP_Lamm_anz: fun_real_int > fun_fu3737244727311182119al_int ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: nat > fun_ar241291507626742315on_nat ).

tff(sy_c_ATP_058Lamm__aoa____,type,
    aTP_Lamm_aoa: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__aob____,type,
    aTP_Lamm_aob: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__aoc____,type,
    aTP_Lamm_aoc: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aod____,type,
    aTP_Lamm_aod: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__aoe____,type,
    aTP_Lamm_aoe: fun_nat_real ).

tff(sy_c_ATP_058Lamm__aof____,type,
    aTP_Lamm_aof: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aog____,type,
    aTP_Lamm_aog: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__aoh____,type,
    aTP_Lamm_aoh: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__aoi____,type,
    aTP_Lamm_aoi: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aoj____,type,
    aTP_Lamm_aoj: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aok____,type,
    aTP_Lamm_aok: fun_nat_real ).

tff(sy_c_ATP_058Lamm__aol____,type,
    aTP_Lamm_aol: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aom____,type,
    aTP_Lamm_aom: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__aon____,type,
    aTP_Lamm_aon: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aoo____,type,
    aTP_Lamm_aoo: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aop____,type,
    aTP_Lamm_aop: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aoq____,type,
    aTP_Lamm_aoq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__aor____,type,
    aTP_Lamm_aor: fun_fu7108391812694488075_nat_o ).

tff(sy_c_ATP_058Lamm__aos____,type,
    aTP_Lamm_aos: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aot____,type,
    aTP_Lamm_aot: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aou____,type,
    aTP_Lamm_aou: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aov____,type,
    aTP_Lamm_aov: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aow____,type,
    aTP_Lamm_aow: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__aox____,type,
    aTP_Lamm_aox: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__aoy____,type,
    aTP_Lamm_aoy: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aoz____,type,
    aTP_Lamm_aoz: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: nat > fun_ar2846716467024420065on_nat ).

tff(sy_c_ATP_058Lamm__apa____,type,
    aTP_Lamm_apa: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__apb____,type,
    aTP_Lamm_apb: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__apc____,type,
    aTP_Lamm_apc: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__apd____,type,
    aTP_Lamm_apd: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ape____,type,
    aTP_Lamm_ape: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__apf____,type,
    aTP_Lamm_apf: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__apg____,type,
    aTP_Lamm_apg: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aph____,type,
    aTP_Lamm_aph: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__api____,type,
    aTP_Lamm_api: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__apj____,type,
    aTP_Lamm_apj: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__apk____,type,
    aTP_Lamm_apk: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__apl____,type,
    aTP_Lamm_apl: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__apm____,type,
    aTP_Lamm_apm: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__apn____,type,
    aTP_Lamm_apn: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__apo____,type,
    aTP_Lamm_apo: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__app____,type,
    aTP_Lamm_app: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__apq____,type,
    aTP_Lamm_apq: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__apr____,type,
    aTP_Lamm_apr: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__aps____,type,
    aTP_Lamm_aps: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__apt____,type,
    aTP_Lamm_apt: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__apu____,type,
    aTP_Lamm_apu: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__apv____,type,
    aTP_Lamm_apv: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__apw____,type,
    aTP_Lamm_apw: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__apx____,type,
    aTP_Lamm_apx: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__apy____,type,
    aTP_Lamm_apy: fun_real_o ).

tff(sy_c_ATP_058Lamm__apz____,type,
    aTP_Lamm_apz: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: nat > fun_ar2846716467024420065on_nat ).

tff(sy_c_ATP_058Lamm__aqa____,type,
    aTP_Lamm_aqa: fun_real_fun_int_o ).

tff(sy_c_ATP_058Lamm__aqb____,type,
    aTP_Lamm_aqb: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__aqc____,type,
    aTP_Lamm_aqc: nat > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__aqd____,type,
    aTP_Lamm_aqd: nat > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__aqe____,type,
    aTP_Lamm_aqe: vEBT_VEBT > fun_VE6841497200278198903t_assn ).

tff(sy_c_ATP_058Lamm__aqf____,type,
    aTP_Lamm_aqf: vEBT_VEBT > fun_VE6841497200278198903t_assn ).

tff(sy_c_ATP_058Lamm__aqg____,type,
    aTP_Lamm_aqg: vEBT_VEBT > fun_VE9201042017749490615o_assn ).

tff(sy_c_ATP_058Lamm__aqh____,type,
    aTP_Lamm_aqh: vEBT_VEBT > fun_VE1035329831157012416o_assn ).

tff(sy_c_ATP_058Lamm__aqi____,type,
    aTP_Lamm_aqi: vEBT_VEBT > fun_VE7792148054706007876t_assn ).

tff(sy_c_ATP_058Lamm__aqj____,type,
    aTP_Lamm_aqj: vEBT_VEBT > fun_VE7792148054706007876t_assn ).

tff(sy_c_ATP_058Lamm__aqk____,type,
    aTP_Lamm_aqk: nat > fun_li3871283220525210964t_assn ).

tff(sy_c_ATP_058Lamm__aql____,type,
    aTP_Lamm_aql: nat > fun_li3871283220525210964t_assn ).

tff(sy_c_ATP_058Lamm__aqm____,type,
    aTP_Lamm_aqm: complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__aqn____,type,
    aTP_Lamm_aqn: assn > fun_Pr6626665806141890376t_assn ).

tff(sy_c_ATP_058Lamm__aqo____,type,
    aTP_Lamm_aqo: list_VEBT_VEBT > fun_ar1453094831611173103i_assn ).

tff(sy_c_ATP_058Lamm__aqp____,type,
    aTP_Lamm_aqp: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__aqq____,type,
    aTP_Lamm_aqq: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__aqr____,type,
    aTP_Lamm_aqr: fun_int_o ).

tff(sy_c_ATP_058Lamm__aqs____,type,
    aTP_Lamm_aqs: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aqt____,type,
    aTP_Lamm_aqt: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aqu____,type,
    aTP_Lamm_aqu: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__aqv____,type,
    aTP_Lamm_aqv: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aqw____,type,
    aTP_Lamm_aqw: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__aqx____,type,
    aTP_Lamm_aqx: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__aqy____,type,
    aTP_Lamm_aqy: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aqz____,type,
    aTP_Lamm_aqz: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: nat > fun_ar2846716467024420065on_nat ).

tff(sy_c_ATP_058Lamm__ara____,type,
    aTP_Lamm_ara: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__arb____,type,
    aTP_Lamm_arb: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__arc____,type,
    aTP_Lamm_arc: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ard____,type,
    aTP_Lamm_ard: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__are____,type,
    aTP_Lamm_are: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__arf____,type,
    aTP_Lamm_arf: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__arg____,type,
    aTP_Lamm_arg: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__arh____,type,
    aTP_Lamm_arh: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ari____,type,
    aTP_Lamm_ari: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__arj____,type,
    aTP_Lamm_arj: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ark____,type,
    aTP_Lamm_ark: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__arl____,type,
    aTP_Lamm_arl: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__arm____,type,
    aTP_Lamm_arm: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__arn____,type,
    aTP_Lamm_arn: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aro____,type,
    aTP_Lamm_aro: fun_Pr6626665806141890376t_assn ).

tff(sy_c_ATP_058Lamm__arp____,type,
    aTP_Lamm_arp: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__arq____,type,
    aTP_Lamm_arq: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__arr____,type,
    aTP_Lamm_arr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ars____,type,
    aTP_Lamm_ars: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__art____,type,
    aTP_Lamm_art: nat > fun_li6914387844523535256_nat_o ).

tff(sy_c_ATP_058Lamm__aru____,type,
    aTP_Lamm_aru: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__arv____,type,
    aTP_Lamm_arv: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__arw____,type,
    aTP_Lamm_arw: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__arx____,type,
    aTP_Lamm_arx: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ary____,type,
    aTP_Lamm_ary: fun_o_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__arz____,type,
    aTP_Lamm_arz: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: nat > fun_ar668040301299616455on_nat ).

tff(sy_c_ATP_058Lamm__asa____,type,
    aTP_Lamm_asa: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__asb____,type,
    aTP_Lamm_asb: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__asc____,type,
    aTP_Lamm_asc: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__asd____,type,
    aTP_Lamm_asd: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ase____,type,
    aTP_Lamm_ase: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__asf____,type,
    aTP_Lamm_asf: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__asg____,type,
    aTP_Lamm_asg: $o > fun_o_5488095450495718970o_char ).

tff(sy_c_ATP_058Lamm__ash____,type,
    aTP_Lamm_ash: $o > fun_o_6360995121869435791o_char ).

tff(sy_c_ATP_058Lamm__asi____,type,
    aTP_Lamm_asi: $o > fun_o_2683029531876921444o_char ).

tff(sy_c_ATP_058Lamm__asj____,type,
    aTP_Lamm_asj: $o > fun_o_3037882179896959673o_char ).

tff(sy_c_ATP_058Lamm__ask____,type,
    aTP_Lamm_ask: $o > fun_o_335877185394996366o_char ).

tff(sy_c_ATP_058Lamm__asl____,type,
    aTP_Lamm_asl: $o > fun_o_2016307092959984611o_char ).

tff(sy_c_ATP_058Lamm__asm____,type,
    aTP_Lamm_asm: $o > fun_Co681828205884031672o_char ).

tff(sy_c_ATP_058Lamm__asn____,type,
    aTP_Lamm_asn: fun_Co681828205884031672o_char ).

tff(sy_c_ATP_058Lamm__aso____,type,
    aTP_Lamm_aso: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__asp____,type,
    aTP_Lamm_asp: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__asq____,type,
    aTP_Lamm_asq: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__asr____,type,
    aTP_Lamm_asr: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__ass____,type,
    aTP_Lamm_ass: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ast____,type,
    aTP_Lamm_ast: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__asu____,type,
    aTP_Lamm_asu: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__asv____,type,
    aTP_Lamm_asv: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__asw____,type,
    aTP_Lamm_asw: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__asx____,type,
    aTP_Lamm_asx: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__asy____,type,
    aTP_Lamm_asy: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__asz____,type,
    aTP_Lamm_asz: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: nat > fun_ar6739270660640029374on_nat ).

tff(sy_c_ATP_058Lamm__ata____,type,
    aTP_Lamm_ata: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__atb____,type,
    aTP_Lamm_atb: fun_real_real ).

tff(sy_c_ATP_058Lamm__atc____,type,
    aTP_Lamm_atc: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__atd____,type,
    aTP_Lamm_atd: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ate____,type,
    aTP_Lamm_ate: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__atf____,type,
    aTP_Lamm_atf: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__atg____,type,
    aTP_Lamm_atg: fun_real_real ).

tff(sy_c_ATP_058Lamm__ath____,type,
    aTP_Lamm_ath: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ati____,type,
    aTP_Lamm_ati: fun_nat_real ).

tff(sy_c_ATP_058Lamm__atj____,type,
    aTP_Lamm_atj: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__atk____,type,
    aTP_Lamm_atk: fun_nat_real ).

tff(sy_c_ATP_058Lamm__atl____,type,
    aTP_Lamm_atl: fun_nat_real ).

tff(sy_c_ATP_058Lamm__atm____,type,
    aTP_Lamm_atm: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__atn____,type,
    aTP_Lamm_atn: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ato____,type,
    aTP_Lamm_ato: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__atp____,type,
    aTP_Lamm_atp: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__atq____,type,
    aTP_Lamm_atq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__atr____,type,
    aTP_Lamm_atr: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ats____,type,
    aTP_Lamm_ats: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__att____,type,
    aTP_Lamm_att: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__atu____,type,
    aTP_Lamm_atu: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__atv____,type,
    aTP_Lamm_atv: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__atw____,type,
    aTP_Lamm_atw: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__atx____,type,
    aTP_Lamm_atx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__aty____,type,
    aTP_Lamm_aty: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__atz____,type,
    aTP_Lamm_atz: fun_real_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: nat > fun_ar125918970492633653on_nat ).

tff(sy_c_ATP_058Lamm__aua____,type,
    aTP_Lamm_aua: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__aub____,type,
    aTP_Lamm_aub: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__auc____,type,
    aTP_Lamm_auc: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__aud____,type,
    aTP_Lamm_aud: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__aue____,type,
    aTP_Lamm_aue: fun_real_real ).

tff(sy_c_ATP_058Lamm__auf____,type,
    aTP_Lamm_auf: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__aug____,type,
    aTP_Lamm_aug: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: option4927543243414619207at_nat > fun_na4765968698818321074on_nat ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: option4927543243414619207at_nat > fun_na4588075804320590989on_nat ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: option4927543243414619207at_nat > fun_na3312191441946882487on_nat ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: option4927543243414619207at_nat > fun_na8209708776965064507on_nat ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: fun_op8229906066235313459T_VEBT ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_o_8142510167905321106T_VEBT ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: option4927543243414619207at_nat > fun_na7845933267320499298on_nat ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_op5783775701855621896on_nat ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: nat > fun_na8204034076925005301on_nat ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: nat > fun_na6142010823788121245on_nat ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: nat > fun_na554821608089334116on_nat ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: nat > fun_na8711077528228054016on_nat ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: nat > fun_na4693688529567471479on_nat ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: nat > fun_na3458768066340930158on_nat ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: nat > fun_op5783775701855621896on_nat ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: nat > fun_o_4049874799390722306on_nat ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: nat > fun_na8307603164994189425on_nat ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: nat > fun_na7745806701448065237on_nat ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: nat > fun_na9055242642580439111on_nat ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: nat > fun_na7511075313506320931on_nat ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: nat > fun_na8752921509028558004on_nat ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: nat > fun_na5280544852190576430on_nat ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: nat > fun_na5280544852190576430on_nat ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: nat > fun_na5280544852190576430on_nat ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: nat > fun_na6850568596825991504on_nat ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: nat > fun_na8435634892762120647on_nat ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: nat > fun_na2272093518330687934on_nat ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: nat > fun_op1686896281538429038on_nat ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: vEBT_VEBT > fun_na8849172711643138838on_nat ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: vEBT_VEBT > fun_na1749477412626277658on_nat ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: vEBT_VEBT > fun_na3054862186897204036on_nat ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: vEBT_VEBT > fun_na5992628573614119403on_nat ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: vEBT_VEBT > fun_na6220387211928018449on_nat ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: set_VEBT_VEBT > fun_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: set_complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: set_Pr958786334691620121nt_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: nat > fun_op3734645406857789385on_nat ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: nat > fun_ar5553829433130746806on_nat ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: nat > fun_ar5028184689208713744on_nat ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: nat > fun_ar3067344226597455191on_nat ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: nat > fun_ar3350189613669807859on_nat ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: nat > fun_na3723828179601053495on_nat ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: nat > fun_na3458768066340930158on_nat ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: nat > fun_op5783775701855621896on_nat ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: nat > fun_o_4049874799390722306on_nat ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_o_o > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_o_o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: fun_o_int > fun_fu6453395667394816515nt_int ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_int_o > fun_fu1481857817119020493_int_o ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_int_int > fun_fu6092417073193843705nt_int ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_Pr4796943543331071514_nat_o > fun_fu3843450905092635034_nat_o ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_o_67843798866261420at_nat > fun_fu4108308376836824364at_nat ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_Pr3238374608701176162_int_o > fun_fu4592178885959468770_int_o ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_Pr982841511892225912nt_int > fun_fu717549361983437348nt_int ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_o_8988054265482332260nt_int > fun_fu8427683764667544036nt_int ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_num_num > fun_fu8155524643419131615um_num ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_num_int > fun_fu726849872193282289um_int ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_int_num > fun_fu5322398707847370225um_num ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_int_int > fun_fu7117095973476296707um_int ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_option_num_num > fun_fu152868422178311599um_num ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_option_num_int > fun_fu1947565687807238081um_int ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_num_option_num > fun_fu4719417816141531311on_num ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_int_option_num > fun_fu7964671872712412353on_num ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_op8989173174992673105on_num > fun_fu2987389051837124479on_num ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_o_o > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_Pr2118491707810114563at_nat > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_Pr4796943543331071514_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_Pr3238374608701176162_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_Pr982841511892225912nt_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_as3983628580203722482n_assn ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_as3983628580203722482n_assn ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_fu6311994054701583983uint32 ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_fu8568212769178095465eger_o ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_fu5407763515825486313uint32 ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: fun_fu713059014528672409uint32 ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: fun_fu5055813439476669081uint32 ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_Pr2118491707810114563at_nat ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: fun_num_num ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: nat > fun_ar7864286482443943364on_nat ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: nat > fun_ar7575598148623397506on_nat ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: nat > fun_ar5515879994638340162on_nat ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: nat > fun_ar4372988750007754556on_nat ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: nat > fun_ar6283851350152303155on_nat ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: nat > fun_ar3874946511942987587on_nat ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: nat > fun_ar3874946511942987587on_nat ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: nat > fun_na4376472396080214220on_nat ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: nat > fun_na5759534986367084828on_nat ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: nat > fun_na7014419191514879763on_nat ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: nat > fun_na2272093518330687934on_nat ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: nat > fun_op1686896281538429038on_nat ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: vEBT_VEBT > fun_na8849172711643138838on_nat ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: vEBT_VEBT > fun_na1749477412626277658on_nat ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: vEBT_VEBT > fun_na3054862186897204036on_nat ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: vEBT_VEBT > fun_na5992628573614119403on_nat ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: vEBT_VEBT > fun_na6220387211928018449on_nat ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: option_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: option_rat > fun_rat_o ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: option_num > fun_num_o ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: option_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: option_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: option_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: option_rat > fun_rat_o ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: option_num > fun_num_o ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: option_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: option_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_Pr7755520105644788814_nat_o ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_Pr3989439477023515934_nat_o ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_num_o ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_nat_o ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_Pr7755520105644788814_nat_o ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_Pr3989439477023515934_nat_o ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_num_o ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_nat_o ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: fun_VE3775134097089128360on_nat > fun_na607092491672936340on_nat ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_VE3775134097089128360on_nat > fun_na4347519868517868204on_nat ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: fun_VE3775134097089128360on_nat > fun_na7760356811779829437on_nat ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_VE3775134097089128360on_nat > fun_na1922817992914042619on_nat ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_VE3775134097089128360on_nat > fun_na1922817992914042619on_nat ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_VE3775134097089128360on_nat > fun_na1922817992914042619on_nat ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_VE3775134097089128360on_nat > fun_na6896436734642855769on_nat ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_VE3775134097089128360on_nat > fun_na56368789070356560on_nat ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_VE3775134097089128360on_nat > fun_na3160842509525366983on_nat ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_VE3775134097089128360on_nat > fun_na7284882202938760507on_nat ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_VE3775134097089128360on_nat > fun_VE5053670003584447325on_nat ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_VE3775134097089128360on_nat > fun_VE205885719767202089on_nat ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_VE3775134097089128360on_nat > fun_VE8661257954422131339on_nat ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: fun_VE3775134097089128360on_nat > fun_VE7418359107739301810on_nat ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_VE3775134097089128360on_nat > fun_VE8881588537313513120on_nat ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_VE3775134097089128360on_nat > fun_na7821417838835476043on_nat ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_VE3775134097089128360on_nat > fun_na4144732571386696713on_nat ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_VE3775134097089128360on_nat > fun_na5199945083401365888on_nat ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_VE3775134097089128360on_nat > fun_na4376472396080214220on_nat ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_VE3775134097089128360on_nat > fun_na4376472396080214220on_nat ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_VE3775134097089128360on_nat > fun_na6803271923218958805on_nat ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_VE3775134097089128360on_nat > fun_na6506281770025666025on_nat ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_VE3775134097089128360on_nat > fun_na223517484589429856on_nat ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_VE3775134097089128360on_nat > fun_na3160842509525366983on_nat ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_VE3775134097089128360on_nat > fun_na7284882202938760507on_nat ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_VE3775134097089128360on_nat > fun_VE5053670003584447325on_nat ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_VE3775134097089128360on_nat > fun_VE205885719767202089on_nat ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_VE3775134097089128360on_nat > fun_VE8661257954422131339on_nat ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_VE3775134097089128360on_nat > fun_VE7418359107739301810on_nat ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_VE3775134097089128360on_nat > fun_VE8881588537313513120on_nat ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_VE8516060933855021849on_nat > fun_na4155721213724604198on_nat ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_VE8516060933855021849on_nat > fun_na1029252574726078829on_nat ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_VE8516060933855021849on_nat > fun_na7271643239918768137on_nat ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: fun_VE8516060933855021849on_nat > fun_na2011684170944547840on_nat ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: fun_VE8516060933855021849on_nat > fun_na2990367117748150135on_nat ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: fun_VE8516060933855021849on_nat > fun_na6220387211928018449on_nat ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_VE8516060933855021849on_nat > fun_na3472305698324104669on_nat ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_VE8516060933855021849on_nat > fun_na505420239896681636on_nat ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_VE8516060933855021849on_nat > fun_na3776232666933163584on_nat ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: fun_VE8516060933855021849on_nat > fun_na3371937797455410308on_nat ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_VE8516060933855021849on_nat > fun_na2990367117748150135on_nat ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_VE8516060933855021849on_nat > fun_na6220387211928018449on_nat ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: $o > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: $o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_na1476630412504255114at_nat > fun_Pr2118491707810114563at_nat ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_na1476630412504255114at_nat > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_nat_o > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_in4211310926729683942nt_int > fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_in4211310926729683942nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: fun_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_int_int > fun_Pr982841511892225912nt_int ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: fun_na1476630412504255114at_nat > fun_Pr2118491707810114563at_nat ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: fun_na1476630412504255114at_nat > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_nat_o > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: fun_in4211310926729683942nt_int > fun_Pr2478283820680522243nt_int ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_in4211310926729683942nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_int_int > fun_Pr982841511892225912nt_int ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_Pr3650993782763339253T_VEBT > fun_fu493888963134383235T_VEBT ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_Pr3650993782763339253T_VEBT > fun_fu4137589202531160609T_VEBT ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_Pr7471953115884902709VEBT_o > fun_fu4430894231949415039VEBT_o ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_Pr7471953115884902709VEBT_o > fun_fu5205589992830698549un_o_o ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_o_3001263932818058983T_VEBT > fun_fu327035696625524243T_VEBT ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_o_3001263932818058983T_VEBT > fun_fu1082429846158404371T_VEBT ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_o_o > fun_fu4591712890356382959VEBT_o ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_o_o > fun_fu4605530428521801667un_o_o ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_op541075054904862521VEBT_o ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_wo7490464530504204009l_num1 ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_real_real ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_int_int ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: fun_uint32_uint32 ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: fun_real_real ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_int_int ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: vEBT_VEBT > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: vEBT_VEBT > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: rat > fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_complex_o ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_real_o ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_int_o ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: set_VEBT_VEBT > fun_se4683600015404540110VEBT_o ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: set_Pr958786334691620121nt_int > fun_se4767462807232413326_int_o ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: fun_He8640631688564546883on_nat > fun_fu8409376595709561667on_nat ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_He8640631688564546883on_nat > fun_fu7388674814006075715on_nat ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_He5988046180898962017Heap_o > fun_fu1930685970884637881Heap_o ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_He5988046180898962017Heap_o > fun_fu2310530280156192909Heap_o ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_He2623223239018661208_VEBTi > fun_fu6559571188541174616_VEBTi ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_He2623223239018661208_VEBTi > fun_fu4739529698945370456_VEBTi ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_He6140748070426431971on_nat > fun_fu2747026234331107907on_nat ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_He6140748070426431971on_nat > fun_fu8544618812552386275on_nat ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_He6597989805181038017Heap_o > fun_fu7474633291259637177Heap_o ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_He6597989805181038017Heap_o > fun_fu144732360711459565Heap_o ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_He6517945810935171576_VEBTi > fun_fu4397613994831067736_VEBTi ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_He6517945810935171576_VEBTi > fun_fu4945982929356410616_VEBTi ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: fun_He1783186893022644846on_nat > fun_fu8147167295154171246on_nat ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_He1783186893022644846on_nat > fun_fu3089959412332122350on_nat ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_He4624006784692404214Heap_o > fun_fu6351278744920802638Heap_o ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_He4624006784692404214Heap_o > fun_fu5088205192217346978Heap_o ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_He8247729123535230979_VEBTi > fun_fu230338336617200387_VEBTi ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_He8247729123535230979_VEBTi > fun_fu6035134072308932227_VEBTi ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: heap_T2636463487746394924on_nat > fun_Pr293967910607515554on_nat ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: heap_Time_Heap_nat > fun_Pr6201033369230528594ap_nat ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: heap_Time_Heap_o > fun_Pr4445936646954502082Heap_o ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: heap_T8145700208782473153_VEBTi > fun_Pr7231383449453139767_VEBTi ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: vEBT_VEBT > fun_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: product_prod_int_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: vEBT_VEBT > fun_se4683600015404540110VEBT_o ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: product_prod_int_int > fun_se4767462807232413326_int_o ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: vEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: product_prod_int_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: vEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: product_prod_int_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: vEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: product_prod_int_int > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_real_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_o_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_int_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_na5542100196373022448T_VEBT > fun_fu3315830660770497029VEBT_o ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: fun_num_option_num > fun_fu2690800231887536361_num_o ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: fun_nat_option_num > fun_fu6438114819929203049_num_o ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_nat_option_nat > fun_fu994687079535449577_nat_o ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_int_option_int > fun_fu7409062071265970153_int_o ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_na5363892732107265798uint32 ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_na4844478536282698896l_num1 ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_VEBT_VEBT_nat > fun_VEBT_VEBT_real ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_nat_nat > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: num > fun_nu5148503272595741012um_num ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: produc4813437837504472865T_VEBT > fun_na1994911354594794003T_VEBT ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: num > fun_na6415222038798377684at_num ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: assn > fun_as229090903528795618n_assn ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: uint32 > fun_uint32_o ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: word_N3645301735248828278l_num1 > fun_wo7859709073469921915num1_o ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: list_VEBT_VEBT > fun_na8277067783256924526Heap_o ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: array_VEBT_VEBTi > fun_li6502840675107701469Heap_o ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: nat > fun_na2926107884465348340Heap_o ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: nat > fun_na1856392443187364084Heap_o ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: nat > fun_na8953503613173268075Heap_o ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: nat > fun_na2536804662393741479Heap_o ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: nat > fun_na8215782167631191514Heap_o ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: nat > fun_na89412549627184089Heap_o ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: nat > fun_op218661150302373393Heap_o ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: nat > fun_op6476899888854096997Heap_o ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: nat > fun_op1410506891405710263Heap_o ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: vEBT_VEBT > fun_na6537123900254388819Heap_o ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: vEBT_VEBT > fun_na7998299765372931779Heap_o ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: vEBT_VEBT > fun_na1841732828589024619Heap_o ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: nat > fun_o_1207901889202581752Heap_o ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: nat > fun_VE7038733415900436564Heap_o ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: array_VEBT_VEBTi > fun_na24089148966465545Heap_o ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: array_VEBT_VEBTi > fun_na1260656955978387516Heap_o ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: nat > fun_na7155415448731535229Heap_o ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: nat > fun_na5646030511792458954Heap_o ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: nat > fun_op1763656958002597022Heap_o ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: fun_VE8181444127144381288Heap_o > fun_li5500215499729237691Heap_o ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: fun_VE8181444127144381288Heap_o > fun_ar8795608905921525346Heap_o ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_VE8181444127144381288Heap_o > fun_na8963100177754871293Heap_o ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: fun_VE8181444127144381288Heap_o > fun_na6370613109607759809Heap_o ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_VE8181444127144381288Heap_o > fun_na2699581809715174900Heap_o ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_VE8181444127144381288Heap_o > fun_na853727736315331060Heap_o ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_VE8181444127144381288Heap_o > fun_na3474800335600542119Heap_o ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: fun_VE8181444127144381288Heap_o > fun_na2622557336161645666Heap_o ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: fun_VE8181444127144381288Heap_o > fun_na7247921347214250394Heap_o ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: fun_VE8181444127144381288Heap_o > fun_na5958411666642256818Heap_o ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_VE8181444127144381288Heap_o > fun_na7955433379684683584Heap_o ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: fun_VE8181444127144381288Heap_o > fun_VE567930495301233690Heap_o ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: fun_VE8181444127144381288Heap_o > fun_VE1467545974346792018Heap_o ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: fun_VE8181444127144381288Heap_o > fun_VE2348196291809386290Heap_o ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: fun_VE7134199479003659425Heap_o > fun_na8576134899853594145Heap_o ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: fun_VE7134199479003659425Heap_o > fun_ar6383509519145165198Heap_o ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: fun_VE7134199479003659425Heap_o > fun_ar8039747222344348737Heap_o ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: fun_VE7134199479003659425Heap_o > fun_na60875040783188486Heap_o ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_VE7134199479003659425Heap_o > fun_na6148071205614320851Heap_o ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: fun_VE7134199479003659425Heap_o > fun_na1841732828589024619Heap_o ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: fun_Pr633315466666964198on_nat > fun_na7821417838835476043on_nat ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: fun_Pr633315466666964198on_nat > fun_na4144732571386696713on_nat ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: fun_Pr633315466666964198on_nat > fun_na5199945083401365888on_nat ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: fun_Pr633315466666964198on_nat > fun_na4376472396080214220on_nat ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: fun_Pr633315466666964198on_nat > fun_na4376472396080214220on_nat ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: fun_Pr633315466666964198on_nat > fun_na6803271923218958805on_nat ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: fun_Pr633315466666964198on_nat > fun_na6506281770025666025on_nat ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: fun_Pr633315466666964198on_nat > fun_na223517484589429856on_nat ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: fun_Pr633315466666964198on_nat > fun_na3160842509525366983on_nat ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: fun_Pr633315466666964198on_nat > fun_na7284882202938760507on_nat ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: fun_Pr633315466666964198on_nat > fun_VE5053670003584447325on_nat ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_Pr633315466666964198on_nat > fun_VE205885719767202089on_nat ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_Pr633315466666964198on_nat > fun_VE8661257954422131339on_nat ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_Pr633315466666964198on_nat > fun_VE7418359107739301810on_nat ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: fun_Pr633315466666964198on_nat > fun_VE8881588537313513120on_nat ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_Pr633315466666964198on_nat > fun_VE3775134097089128360on_nat ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: produc3960855945107176009Ti_nat > fun_fu8904838144393377183on_nat ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_Pr633315466666964198on_nat > fun_na607092491672936340on_nat ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: fun_Pr633315466666964198on_nat > fun_na4347519868517868204on_nat ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_Pr633315466666964198on_nat > fun_na7760356811779829437on_nat ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: fun_Pr633315466666964198on_nat > fun_na1922817992914042619on_nat ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: fun_Pr633315466666964198on_nat > fun_na1922817992914042619on_nat ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_Pr633315466666964198on_nat > fun_na1922817992914042619on_nat ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_Pr633315466666964198on_nat > fun_na6896436734642855769on_nat ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_Pr633315466666964198on_nat > fun_na56368789070356560on_nat ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_Pr633315466666964198on_nat > fun_na3160842509525366983on_nat ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_Pr633315466666964198on_nat > fun_na7284882202938760507on_nat ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_Pr633315466666964198on_nat > fun_VE5053670003584447325on_nat ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: fun_Pr633315466666964198on_nat > fun_VE205885719767202089on_nat ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_Pr633315466666964198on_nat > fun_VE8661257954422131339on_nat ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: fun_Pr633315466666964198on_nat > fun_VE7418359107739301810on_nat ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: fun_Pr633315466666964198on_nat > fun_VE8881588537313513120on_nat ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_Pr633315466666964198on_nat > fun_VE3775134097089128360on_nat ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: produc3960855945107176009Ti_nat > fun_fu8904838144393377183on_nat ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_o_2238389867833408620_VEBTi > fun_fu1246984265994403236_VEBTi ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: fun_na7848093160509028170_VEBTi > fun_fu9134463466825115560_VEBTi ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: fun_na7848093160509028170_VEBTi > fun_fu7298712310433895232Heap_o ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: fun_na2343073368905879023Heap_o > fun_fu1560271115773023055Heap_o ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_o_8325599348076409421Heap_o > fun_fu7818770860635125077_VEBTi ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: fun_na2343073368905879023Heap_o > fun_fu4435269841469898295_VEBTi ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: fun_o_4158591058828729991ap_nat > fun_fu2016835990695057225_VEBTi ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: fun_na3533371890617757541ap_nat > fun_fu6094620073291891011_VEBTi ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: fun_na3533371890617757541ap_nat > fun_fu7046341792740000859Heap_o ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_li7499208429157053461_VEBTi > fun_fu1491295905708448371_VEBTi ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: fun_na5845961453008041334_VEBTi > fun_fu6710363165572686898_VEBTi ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_fu5786974620624140868ap_nat > fun_fu1693797150997279004_VEBTi ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: fun_VE6491116402277797979_VEBTi > fun_fu2110818295914825613_VEBTi ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_fu4424605311360796457_VEBTi > fun_fu4571548849563991809_VEBTi ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: fun_na8517577583218006873Heap_o > fun_fu7058568179839949213Heap_o ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_fu5786974620624140868ap_nat > fun_fu2518310113503212154Heap_o ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: fun_VE6943415934277719092Heap_o > fun_fu4172174496237376130Heap_o ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: fun_fu4424605311360796457_VEBTi > fun_fu8892947379916810463Heap_o ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: fun_o_7171189317104958676_VEBTi > fun_fu5700640950803245722_VEBTi ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_fu335202653675969872Heap_o > fun_fu6875650659019086864_VEBTi ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_o_2223814739247024763Heap_o > fun_fu7077623433360679221Heap_o ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: fun_fu335202653675969872Heap_o > fun_fu4588794411785514862Heap_o ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: fun_na979242554834862325Heap_o > fun_fu7978248880665362293Heap_o ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: fun_fu2606674006500944492ap_nat > fun_fu1468149900866469658Heap_o ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_na7267038673063590618_VEBTi > fun_fu2750827917295066202_VEBTi ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: fun_fu2606674006500944492ap_nat > fun_fu5185620903687836730_VEBTi ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: fun_o_5993975905899854163Heap_o > fun_fu2648038516759057757Heap_o ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: fun_fu7983188961739674408Heap_o > fun_fu12881409451275790Heap_o ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: fun_o_2166024663187074044_VEBTi > fun_fu612931627712977010_VEBTi ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_fu7983188961739674408Heap_o > fun_fu2729612641518205230_VEBTi ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: heap_T2636463487746394924on_nat > fun_fu4400511211004959420on_nat ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: heap_Time_Heap_o > fun_fu335202653675969872Heap_o ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: heap_T8145700208782473153_VEBTi > fun_fu6714510707232648287_VEBTi ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: heap_T2636463487746394924on_nat > fun_fu8904838144393377183on_nat ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: heap_Time_Heap_o > fun_fu7243727008143648435Heap_o ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: fun_na5556145776640407093on_nat > fun_Pr5956696566359781750on_nat ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_na2343073368905879023Heap_o > fun_Pr4877903221429077380Heap_o ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: heap_T8145700208782473153_VEBTi > fun_Pr4404027214751104635_VEBTi ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: heap_T2636463487746394924on_nat > fun_Pr633315466666964198on_nat ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: heap_Time_Heap_o > fun_Pr940977554174145406Heap_o ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_fu6711510597495717193_nat_o > fun_fu7108391812694488075_nat_o ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_he1353482274668092617it_nat ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_VE3775134097089128360on_nat ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_VE3775134097089128360on_nat > fun_VE3775134097089128360on_nat ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: fun_VE3775134097089128360on_nat > fun_VE3775134097089128360on_nat ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_Pr342922516890243977on_nat > fun_na3472305698324104669on_nat ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_Pr342922516890243977on_nat > fun_na505420239896681636on_nat ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: fun_Pr342922516890243977on_nat > fun_na3776232666933163584on_nat ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: fun_Pr342922516890243977on_nat > fun_na3371937797455410308on_nat ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: fun_Pr342922516890243977on_nat > fun_na2990367117748150135on_nat ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: fun_Pr342922516890243977on_nat > fun_na6220387211928018449on_nat ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: fun_Pr342922516890243977on_nat > fun_VE8516060933855021849on_nat ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: produc3881548065746020326Ti_nat > fun_fu4400511211004959420on_nat ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: fun_Pr342922516890243977on_nat > fun_na4155721213724604198on_nat ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_Pr342922516890243977on_nat > fun_na1029252574726078829on_nat ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: fun_Pr342922516890243977on_nat > fun_na7271643239918768137on_nat ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: fun_Pr342922516890243977on_nat > fun_na2011684170944547840on_nat ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: fun_Pr342922516890243977on_nat > fun_na2990367117748150135on_nat ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: fun_Pr342922516890243977on_nat > fun_na6220387211928018449on_nat ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: fun_Pr342922516890243977on_nat > fun_VE8516060933855021849on_nat ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: produc3881548065746020326Ti_nat > fun_fu4400511211004959420on_nat ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: set_Pr8218934625190621173um_num > fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: set_Pr563407847431865468T_VEBT > fun_na5540905182366533063VEBT_o ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: set_Pr6200539531224447659at_num > fun_nat_fun_num_o ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: set_Pr958786334691620121nt_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_he1183298388918465219it_nat ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_he3045699446853741534it_nat ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_he4386021626318620153it_nat ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: fun_Pr238758252084143515Heap_o > fun_na8576134899853594145Heap_o ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: fun_Pr238758252084143515Heap_o > fun_ar6383509519145165198Heap_o ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: fun_Pr238758252084143515Heap_o > fun_ar8039747222344348737Heap_o ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: fun_Pr238758252084143515Heap_o > fun_na60875040783188486Heap_o ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: fun_Pr238758252084143515Heap_o > fun_na6148071205614320851Heap_o ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: fun_Pr238758252084143515Heap_o > fun_na1841732828589024619Heap_o ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_Pr238758252084143515Heap_o > fun_VE7134199479003659425Heap_o ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: produc3881548065746020326Ti_nat > fun_fu335202653675969872Heap_o ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_Pr940977554174145406Heap_o > fun_li5500215499729237691Heap_o ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: fun_Pr940977554174145406Heap_o > fun_ar8795608905921525346Heap_o ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: fun_Pr940977554174145406Heap_o > fun_na8963100177754871293Heap_o ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: fun_Pr940977554174145406Heap_o > fun_na6370613109607759809Heap_o ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: fun_Pr940977554174145406Heap_o > fun_na2699581809715174900Heap_o ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_Pr940977554174145406Heap_o > fun_na853727736315331060Heap_o ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_Pr940977554174145406Heap_o > fun_na3474800335600542119Heap_o ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: fun_Pr940977554174145406Heap_o > fun_na2622557336161645666Heap_o ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_Pr940977554174145406Heap_o > fun_na7247921347214250394Heap_o ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: fun_Pr940977554174145406Heap_o > fun_na5958411666642256818Heap_o ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_Pr940977554174145406Heap_o > fun_na7955433379684683584Heap_o ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_Pr940977554174145406Heap_o > fun_VE567930495301233690Heap_o ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: fun_Pr940977554174145406Heap_o > fun_VE1467545974346792018Heap_o ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: fun_Pr940977554174145406Heap_o > fun_VE2348196291809386290Heap_o ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: fun_Pr940977554174145406Heap_o > fun_VE8181444127144381288Heap_o ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: produc3960855945107176009Ti_nat > fun_fu7243727008143648435Heap_o ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: fun_fu4750890421531948930_nat_o > fun_fu1312537534749845416_nat_o ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_VE8516060933855021849on_nat ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_VE8516060933855021849on_nat > fun_VE8516060933855021849on_nat ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_VE8516060933855021849on_nat > fun_VE8516060933855021849on_nat ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_fu2707964056798801139ap_o_o > fun_fu8879972437492397021ap_o_o ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_VE8181444127144381288Heap_o ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: fun_VE8181444127144381288Heap_o > fun_VE8181444127144381288Heap_o ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_nat_int ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_fu8886734894937297424ap_o_o > fun_fu9110254849647837056ap_o_o ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_VE7134199479003659425Heap_o ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: fun_VE7134199479003659425Heap_o > fun_VE7134199479003659425Heap_o ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_nat_real ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: fun_fu411293875173739919on_nat > fun_Pr6807451334997518105on_nat ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: fun_fu2426956696497423235Heap_o > fun_Pr5859542034317231789Heap_o ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: fun_fu1151686773700218227_VEBTi > fun_Pr7000814954542638233_VEBTi ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: fun_fu3620910142932681on_nat > fun_Pr7105669390387666649on_nat ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: fun_fu8308473583445096899Heap_o > fun_Pr2479799759690407533Heap_o ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_nat_complex > fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: fun_nat_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: fun_fu1421233379579251956EBTi_o > fun_fu424623322819101878EBTi_o ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_VE2018681615261037117_VEBTi ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: nat > fun_na2880281091838646604_VEBTi ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: nat > fun_na4683941076921484125_VEBTi ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: fun_VE2018681615261037117_VEBTi > fun_na2548587511393692655_VEBTi ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: fun_VE2018681615261037117_VEBTi > fun_na7679953648084048589_VEBTi ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: fun_VE2018681615261037117_VEBTi > fun_na2840261909164627793_VEBTi ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: fun_VE2018681615261037117_VEBTi > fun_na7137134861304782383_VEBTi ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: fun_VE2018681615261037117_VEBTi > fun_na7565023553309985968_VEBTi ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: fun_VE2018681615261037117_VEBTi > fun_na1347330265571094344_VEBTi ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: fun_VE2018681615261037117_VEBTi > fun_na1347330265571094344_VEBTi ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: fun_VE2018681615261037117_VEBTi > fun_na1659118370164772413_VEBTi ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: fun_VE2018681615261037117_VEBTi > fun_na2169200058431587252_VEBTi ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: fun_VE2018681615261037117_VEBTi > fun_na719414919122706795_VEBTi ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: fun_VE2018681615261037117_VEBTi > fun_na9095497340641005739_VEBTi ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: fun_VE2018681615261037117_VEBTi > fun_na4748561171398030626_VEBTi ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: fun_VE2018681615261037117_VEBTi > fun_na817248180520022297_VEBTi ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: fun_VE2018681615261037117_VEBTi > fun_na5699700385611646096_VEBTi ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: fun_VE2018681615261037117_VEBTi > fun_na113775696040589571_VEBTi ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: fun_VE2018681615261037117_VEBTi > fun_na113775696040589571_VEBTi ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: fun_VE2018681615261037117_VEBTi > fun_na653084535925934713_VEBTi ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: fun_VE2018681615261037117_VEBTi > fun_na993102950844727003_VEBTi ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: fun_VE2018681615261037117_VEBTi > fun_na4691356797839745173_VEBTi ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: fun_VE2018681615261037117_VEBTi > fun_na7353129812761597833_VEBTi ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: fun_VE2018681615261037117_VEBTi > fun_VE8810447512154760383_VEBTi ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: fun_VE2018681615261037117_VEBTi > fun_VE91144239014113491_VEBTi ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: fun_VE2018681615261037117_VEBTi > fun_VE728358326801380661_VEBTi ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: nat > fun_o_6972675622308511895_VEBTi ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: fun_VE2018681615261037117_VEBTi > fun_VE2018681615261037117_VEBTi ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: fun_Pr4171499318505651870_VEBTi > fun_na2890314705291535822_VEBTi ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: fun_Pr4171499318505651870_VEBTi > fun_na7797079703196089028_VEBTi ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: fun_Pr4171499318505651870_VEBTi > fun_na5084500599656708831_VEBTi ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: fun_Pr4171499318505651870_VEBTi > fun_na9009963084909443284_VEBTi ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: fun_Pr4171499318505651870_VEBTi > fun_na2291395713665613775_VEBTi ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: fun_Pr4171499318505651870_VEBTi > fun_na7682395145315030854_VEBTi ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: fun_Pr4171499318505651870_VEBTi > fun_na7174282198332753469_VEBTi ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: fun_Pr4171499318505651870_VEBTi > fun_na3224852944424307892_VEBTi ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: fun_Pr4171499318505651870_VEBTi > fun_na460406568315038379_VEBTi ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: fun_Pr4171499318505651870_VEBTi > fun_na2086927332677804066_VEBTi ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: fun_Pr4171499318505651870_VEBTi > fun_na2740539754339541653_VEBTi ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: fun_Pr4171499318505651870_VEBTi > fun_na278150576929200012_VEBTi ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: fun_Pr4171499318505651870_VEBTi > fun_na8762902223149059366_VEBTi ).

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tff(sy_c_ATP_058Lamm__yv____,type,
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tff(sy_c_ATP_058Lamm__zf____,type,
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tff(sy_c_ATP_058Lamm__zj____,type,
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tff(sy_c_ATP_058Lamm__zk____,type,
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tff(sy_c_ATP_058Lamm__zl____,type,
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tff(sy_c_ATP_058Lamm__zm____,type,
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tff(sy_c_ATP_058Lamm__zn____,type,
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tff(sy_c_ATP_058Lamm__zp____,type,
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tff(sy_c_ATP_058Lamm__zt____,type,
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tff(sy_c_ATP_058Lamm__zu____,type,
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tff(sy_c_ATP_058Lamm__zv____,type,
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tff(sy_c_ATP_058Lamm__zw____,type,
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tff(sy_c_ATP_058Lamm__zx____,type,
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tff(sy_c_ATP_058Lamm__zy____,type,
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tff(sy_c_ATP_058Lamm__zz____,type,
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tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
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tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
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tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
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tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
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tff(sy_c_Array__Time_Olen_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Array__Time_Onth_001_Eo,type,
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tff(sy_c_Array__Time_Onth_001t__Nat__Onat,type,
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tff(sy_c_Array__Time_Onth_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_Array__Time_Oof__list_001_Eo,type,
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tff(sy_c_Array__Time_Oof__list_001t__Int__Oint,type,
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tff(sy_c_Array__Time_Oof__list_001t__Nat__Onat,type,
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tff(sy_c_Array__Time_Oof__list_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Array__Time_Oupd_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Assertions_Oassn_ORep__assn,type,
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tff(sy_c_Assertions_Oentails,type,
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tff(sy_c_Assertions_Opure__assn,type,
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tff(sy_c_Automation_OFI,type,
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tff(sy_c_Binomial_Obinomial,type,
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tff(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits_001t__Int__Oint,type,
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tff(sy_c_Bit__Comprehension_Owf__set__bits__int,type,
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tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Bit__Operations_Oand__not__num,type,
    bit_and_not_num: num > fun_num_option_num ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: nat > fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
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    bit_se5742574853984576102uint32: fun_na3857091055356329634uint32 ).

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    bit_se837345729053750000l_num1: ( nat * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se2793503036327961859nteger: nat > fun_Co8499238673658658103nteger ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
    bit_se7879613467334960850it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
    bit_se7882103937844011126it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Uint32__Ouint32,type,
    bit_se6647067497041451410uint32: ( nat * uint32 ) > uint32 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Word__Oword_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,
    bit_se4894374433684937756l_num1: ( nat * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
    bit_se2923211474154528505it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
    bit_se2925701944663578781it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se8260200283734997820nteger: nat > fun_Co8499238673658658103nteger ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
    bit_se4203085406695923979it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
    bit_se4205575877204974255it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Uint32__Ouint32,type,
    bit_se4315839071623982667uint32: ( nat * uint32 ) > uint32 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Word__Oword_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,
    bit_se5331074070815623765l_num1: ( nat * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Code____Numeral__Ointeger,type,
    bit_se3222712562003087583nteger: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
    bit_se6526347334894502574or_int: fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
    bit_se6528837805403552850or_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Code____Numeral__Ointeger,type,
    bit_se9216721137139052372nteger: fun_Co1375048237525739357_nat_o ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
    bit_se1146084159140164899it_int: fun_int_fun_nat_o ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: fun_nat_fun_nat_o ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Uint32__Ouint32,type,
    bit_se5367290876889521763uint32: fun_uint32_fun_nat_o ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Word__Oword_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,
    bit_se6859397288646540909l_num1: word_N3645301735248828278l_num1 > fun_nat_o ).

tff(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: nat > fun_num_option_num ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl_001t__Nat__Onat,type,
    bit_Sh3965577149348748681tl_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftr_001t__Nat__Onat,type,
    bit_Sh2154871086232339855tr_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bits__Integer_Obin__last__integer,type,
    bits_b8758750999018896077nteger: fun_Code_integer_o ).

tff(sy_c_Bits__Integer_Obin__rest__integer,type,
    bits_b2549910563261871055nteger: fun_Co8499238673658658103nteger ).

tff(sy_c_Bits__Integer_Ointeger__set__bit,type,
    bits_integer_set_bit: ( code_integer * code_integer * $o ) > code_integer ).

tff(sy_c_Bits__Integer_Ointeger__shiftl,type,
    bits_integer_shiftl: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_Bits__Integer_Ointeger__shiftr,type,
    bits_integer_shiftr: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_Code__Int__Integer__Conversion_Oint__of__integer__symbolic,type,
    code_I935103866777955880mbolic: fun_Code_integer_int ).

tff(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).

tff(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: ( code_integer * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: ( code_integer * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_Code__Numeral_Odup,type,
    code_dup: fun_Co8499238673658658103nteger ).

tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: fun_Code_integer_int ).

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: fun_int_Code_integer ).

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

tff(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: fun_Code_integer_nat ).

tff(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: fun_Code_integer_num ).

tff(sy_c_Code__Target__Int_Onegative,type,
    code_Target_negative: fun_num_int ).

tff(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Opartial__term__of__sample_001t__Int__Oint,type,
    code_T3171777517611823113le_int: ( fun_Co4113693717567439989nt_int * int ) > fun_Code_integer_int ).

tff(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Opartial__term__of__sample_001t__Nat__Onat,type,
    code_T3174267988120873389le_nat: ( fun_Co4416855287806144957at_nat * nat ) > fun_Code_integer_nat ).

tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
    comple5871978649864577227Heap_o: ( fun_se3420632608416671075Heap_o * fun_fu6850233865993928942ap_o_o * fun_fu8879972437492397021ap_o_o ) > $o ).

tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
    comple5664458124825824345on_nat: ( fun_se1756668854628305023on_nat * fun_fu5284765496190745342_nat_o * fun_fu7108391812694488075_nat_o ) > $o ).

tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    comple8556566605269262190_VEBTi: ( fun_se3949293115670521129_VEBTi * fun_fu3145514261243071828EBTi_o * fun_fu424623322819101878EBTi_o ) > $o ).

tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
    comple8003709893544916456Heap_o: ( fun_se7475304312357925539Heap_o * fun_fu2183070276626753582ap_o_o * fun_fu9110254849647837056ap_o_o ) > $o ).

tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
    comple7368800650122905596on_nat: ( fun_se772746884452211909on_nat * fun_fu7962274784768186808_nat_o * fun_fu1312537534749845416_nat_o ) > $o ).

tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    comple7878442886591315089_VEBTi: ( fun_se5764602393054361839_VEBTi * fun_fu175107047954615438EBTi_o * fun_fu8912552214369285843EBTi_o ) > $o ).

tff(sy_c_Complete__Partial__Order_Occpo_Ofixp_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
    comple5362527178602408641Heap_o: ( fun_se3420632608416671075Heap_o * fun_fu6850233865993928942ap_o_o * fun_fu8308473583445096899Heap_o ) > fun_Pr940977554174145406Heap_o ).

tff(sy_c_Complete__Partial__Order_Occpo_Ofixp_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
    comple4104493882897929443on_nat: ( fun_se1756668854628305023on_nat * fun_fu5284765496190745342_nat_o * fun_fu3620910142932681on_nat ) > fun_Pr633315466666964198on_nat ).

tff(sy_c_Complete__Partial__Order_Occpo_Ofixp_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    comple5861331542469577976_VEBTi: ( fun_se3949293115670521129_VEBTi * fun_fu3145514261243071828EBTi_o * fun_fu1151686773700218227_VEBTi ) > fun_Pr4404027214751104635_VEBTi ).

tff(sy_c_Complete__Partial__Order_Occpo_Ofixp_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
    comple8693199471913871838Heap_o: ( fun_se7475304312357925539Heap_o * fun_fu2183070276626753582ap_o_o * fun_fu2426956696497423235Heap_o ) > fun_Pr238758252084143515Heap_o ).

tff(sy_c_Complete__Partial__Order_Occpo_Ofixp_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
    comple6035428378238101126on_nat: ( fun_se772746884452211909on_nat * fun_fu7962274784768186808_nat_o * fun_fu411293875173739919on_nat ) > fun_Pr342922516890243977on_nat ).

tff(sy_c_Complete__Partial__Order_Omonotone_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    comple5241396456474302457Heap_o: ( fun_fu6850233865993928942ap_o_o * fun_He2061078463025973126ap_o_o * fun_fu7243727008143648435Heap_o ) > $o ).

tff(sy_c_Complete__Partial__Order_Omonotone_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    comple4513117127444775325on_nat: ( fun_fu5284765496190745342_nat_o * fun_He5905523610488218222_nat_o * fun_fu8904838144393377183on_nat ) > $o ).

tff(sy_c_Complete__Partial__Order_Omonotone_001t__fun_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    comple422786390015765981_VEBTi: ( fun_fu3145514261243071828EBTi_o * fun_He5596775886410745902EBTi_o * fun_fu6714510707232648287_VEBTi ) > $o ).

tff(sy_c_Complete__Partial__Order_Omonotone_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    comple6933591415923782038Heap_o: ( fun_fu2183070276626753582ap_o_o * fun_He2061078463025973126ap_o_o * fun_fu335202653675969872Heap_o ) > $o ).

tff(sy_c_Complete__Partial__Order_Omonotone_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    comple3617025000840655848ap_nat: ( fun_fu2183070276626753582ap_o_o * fun_He1575238268858778734_nat_o * fun_fu5786974620624140868ap_nat ) > $o ).

tff(sy_c_Complete__Partial__Order_Omonotone_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    comple7548324300139837517_VEBTi: ( fun_fu2183070276626753582ap_o_o * fun_He5596775886410745902EBTi_o * fun_fu4424605311360796457_VEBTi ) > $o ).

tff(sy_c_Complete__Partial__Order_Omonotone_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    comple3937103672976186900Heap_o: ( fun_fu7962274784768186808_nat_o * fun_He2061078463025973126ap_o_o * fun_fu7983188961739674408Heap_o ) > $o ).

tff(sy_c_Complete__Partial__Order_Omonotone_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    comple6879160010309051114ap_nat: ( fun_fu7962274784768186808_nat_o * fun_He1575238268858778734_nat_o * fun_fu2606674006500944492ap_nat ) > $o ).

tff(sy_c_Complete__Partial__Order_Omonotone_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    comple7427593846244870586on_nat: ( fun_fu7962274784768186808_nat_o * fun_He5905523610488218222_nat_o * fun_fu4400511211004959420on_nat ) > $o ).

tff(sy_c_Complete__Partial__Order_Omonotone_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    comple2136407771590434127_VEBTi: ( fun_fu7962274784768186808_nat_o * fun_He5596775886410745902EBTi_o * fun_fu9179568866395467601_VEBTi ) > $o ).

tff(sy_c_Complete__Partial__Order_Omonotone_001t__fun_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    comple6664421360520557306_VEBTi: ( fun_fu175107047954615438EBTi_o * fun_He5596775886410745902EBTi_o * fun_fu6595418844270726780_VEBTi ) > $o ).

tff(sy_c_Complex_OArg,type,
    arg: fun_complex_real ).

tff(sy_c_Complex_Ocis,type,
    cis: fun_real_complex ).

tff(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: real > fun_real_complex ).

tff(sy_c_Complex_Ocsqrt,type,
    csqrt: fun_complex_complex ).

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

tff(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
    has_fi5821293074295781190e_real: ( fun_real_real * real * filter_real ) > $o ).

tff(sy_c_Divides_Oadjust__div,type,
    adjust_div: fun_Pr982841511892225912nt_int ).

tff(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: fun_int_fun_int_int ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: fun_na4138050610385705875at_nat ).

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int ) > fun_Pr3238374608701176162_int_o ).

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

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
    unique6319869463603278526ux_int: fun_Pr3238374608701176162_int_o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
    unique6322359934112328802ux_nat: fun_Pr4796943543331071514_nat_o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
    unique3479559517661332726nteger: ( num * num ) > produc8923325533196201883nteger ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
    unique5052692396658037445od_int: ( num * num ) > product_prod_int_int ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
    unique5055182867167087721od_nat: ( num * num ) > product_prod_nat_nat ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
    unique4921790084139445826nteger: ( num * produc8923325533196201883nteger ) > produc8923325533196201883nteger ).

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tff(sy_c_List_Olist_Omap_001t__Real__Oreal_001_Eo,type,
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tff(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_List_Olist_Oset_001_Eo,type,
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tff(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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    upto_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Misc_Orel__of_001t__Int__Oint_001t__Int__Oint,type,
    rel_of_int_int: ( fun_int_option_int * fun_Pr3238374608701176162_int_o ) > set_Pr958786334691620121nt_int ).

tff(sy_c_Misc_Orel__of_001t__Nat__Onat_001t__Nat__Onat,type,
    rel_of_nat_nat: ( fun_nat_option_nat * fun_Pr4796943543331071514_nat_o ) > set_Pr1261947904930325089at_nat ).

tff(sy_c_Misc_Orel__of_001t__Nat__Onat_001t__Num__Onum,type,
    rel_of_nat_num: ( fun_nat_option_num * fun_Pr5555124106300582480_num_o ) > set_Pr6200539531224447659at_num ).

tff(sy_c_Misc_Orel__of_001t__Nat__Onat_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J,type,
    rel_of7001861737123029207T_VEBT: ( fun_na5542100196373022448T_VEBT * fun_Pr6320724876807202603VEBT_o ) > set_Pr563407847431865468T_VEBT ).

tff(sy_c_Misc_Orel__of_001t__Num__Onum_001t__Num__Onum,type,
    rel_of_num_num: ( fun_num_option_num * fun_Pr6331313853612757382_num_o ) > set_Pr8218934625190621173um_num ).

tff(sy_c_Most__significant__bit_Omsb__class_Omsb_001t__Code____Numeral__Ointeger,type,
    most_s5661112943643946085nteger: fun_Code_integer_o ).

tff(sy_c_Most__significant__bit_Omsb__class_Omsb_001t__Int__Oint,type,
    most_s5051101344085556sb_int: fun_int_o ).

tff(sy_c_Most__significant__bit_Omsb__class_Omsb_001t__Uint32__Ouint32,type,
    most_s9063628576841037300uint32: fun_uint32_o ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

tff(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
    case_nat_o: ( $o * fun_nat_o ) > fun_nat_o ).

tff(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
    case_nat_nat: ( nat * fun_nat_nat ) > fun_nat_nat ).

tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
    case_nat_option_num: ( option_num * fun_nat_option_num ) > fun_nat_option_num ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: fun_nat_Code_integer ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: fun_nat_complex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
    semiri4216267220026989637d_enat: nat > extended_enat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Uint32__Ouint32,type,
    semiri2565882477558803405uint32: fun_nat_uint32 ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Word__Oword_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,
    semiri8819519690708144855l_num1: fun_na2984781812408219459l_num1 ).

tff(sy_c_Nat_Osize__class_Osize_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    size_s2700093152935483318Heap_o: heap_Time_Heap_o > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    size_s6287829766004316056on_nat: heap_T2636463487746394924on_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    size_s8425857057747876397_VEBTi: heap_T8145700208782473153_VEBTi > nat ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Assertions__Oassn_J,type,
    size_size_list_assn: list_assn > nat ).

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

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

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
    size_size_list_num: list_num > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
    size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
    size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).

tff(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 ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    size_s6829681357464350627n_assn: list_P8527749157015355191n_assn > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s4764337671732037139T_VEBT: list_P6730324909620535719T_VEBT > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
    size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    size_s7910714270633306959t_real: list_P3644420460460130531t_real > nat ).

tff(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 ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J_J,type,
    size_s8610625264895183403al_int: list_P4344331454722006975al_int > nat ).

tff(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 ).

tff(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 ).

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

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
    size_size_num: fun_num_nat ).

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

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
    size_size_option_num: option_num > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    size_s364044314319911927it_nat: option7339022715339332451it_nat > nat ).

tff(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 ).

tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    size_s3991424295186984831it_nat: option2621746655072343315it_nat > nat ).

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__VEBT____BuildupMemImp__OVEBTi,type,
    size_size_VEBT_VEBTi: fun_VEBT_VEBTi_nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
    size_size_VEBT_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__Word__Oword_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,
    size_s8261804613246490634l_num1: word_N3645301735248828278l_num1 > nat ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: fun_nat_set_nat ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

tff(sy_c_Num_OBitM,type,
    bitM: fun_num_num ).

tff(sy_c_Num_Oinc,type,
    inc: fun_num_num ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
    neg_nu7009210354673126013omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: fun_int_int ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
    neg_numeral_dbl_real: fun_real_real ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Word__Oword_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,
    neg_nu7865238048354675525l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
    neg_nu6511756317524482435omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
    neg_nu3179335615603231917ec_rat: rat > rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
    neg_nu6075765906172075777c_real: fun_real_real ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Uint32__Ouint32,type,
    neg_nu965353292909893953uint32: uint32 > uint32 ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Word__Oword_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,
    neg_nu93272222329896523l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun_num_num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

tff(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
    case_num_option_num: ( option_num * fun_num_option_num * fun_num_option_num ) > fun_num_option_num ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: fun_num_nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: fun_nat_num ).

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

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

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

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: fun_num_int ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: fun_num_nat ).

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

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

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

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Word__Oword_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,
    numera7442385471795722001l_num1: num > word_N3645301735248828278l_num1 ).

tff(sy_c_Num_Opow,type,
    pow: fun_num_fun_num_num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: fun_num_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P7668321371905463026it_nat: option7339022715339332451it_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    none_P2377608414092835994nt_int: option4624381673175914239nt_int ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    none_P6264349658649815852at_num: option642762832853965969at_num ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P281974696781278558it_nat: option7211493094183709123it_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J,type,
    none_P330983522480640549T_VEBT: option254855292876462358T_VEBT ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    none_P4394680061957285238um_num: option2661157926820139483um_num ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P1551326421579882414it_nat: option2621746655072343315it_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P7832717587476222275it_nat: option5408194888911472936it_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
    some_int: fun_int_option_int ).

tff(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: fun_nat_option_nat ).

tff(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
    some_num: fun_num_option_num ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    some_P468703482102919278it_nat: produc8047831477865546771it_nat > option7339022715339332451it_nat ).

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

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J,type,
    some_P2407225848856114310T_VEBT: produc4813437837504472865T_VEBT > option1280308654545898343T_VEBT ).

tff(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 ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    some_P8071634352977444016at_num: product_prod_nat_num > option642762832853965969at_num ).

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Option_Ooption_OSome_001t__Rat__Orat,type,
    some_rat: rat > option_rat ).

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

tff(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Nat__Onat_J,type,
    some_set_nat: set_nat > option_set_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Int__Oint,type,
    case_option_o_int: ( $o * fun_int_o * option_int ) > $o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Nat__Onat,type,
    case_option_o_nat: ( $o * fun_nat_o ) > fun_option_nat_o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Num__Onum,type,
    case_option_o_num: ( $o * fun_num_o ) > fun_option_num_o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o1358941076187788256it_nat: ( $o * fun_Pr3989439477023515934_nat_o ) > fun_op7459662026604166478_nat_o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o184042715313410164at_nat: ( $o * fun_Pr4796943543331071514_nat_o ) > fun_op8028897214545316308_nat_o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o535201446637900608it_nat: ( $o * fun_Pr7755520105644788814_nat_o ) > fun_op2107615217426680968_nat_o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Rat__Orat,type,
    case_option_o_rat: ( $o * fun_rat_o * option_rat ) > $o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Real__Oreal,type,
    case_option_o_real: ( $o * fun_real_o * option_real ) > $o ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Option_Ooption_Ocase__option_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_J_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_J,type,
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tff(sy_c_Option_Ooption_Ocase__option_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
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tff(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Othe_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
    bot_bo3657453288633029378_num_o: fun_num_fun_num_o ).

tff(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
    ord_Least_nat: fun_nat_o > nat ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
    ord_le6747313008572928689nteger: fun_Co2081214437782444048eger_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
    ord_le72135733267957522d_enat: extended_enat > fun_Extended_enat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
    ord_less_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
    ord_less_num: fun_num_fun_num_o ).

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

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_less_option_nat: fun_op5794009454290973322_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Num__Onum_J,type,
    ord_less_option_num: fun_op493924057549976310_num_o ).

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

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

tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
    ord_less_real: fun_real_fun_real_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le1307284697595431911nteger: set_Code_integer > fun_se7289222521248229150eger_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_less_set_complex: set_complex > fun_set_complex_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_set_int: set_int > fun_set_int_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > fun_set_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
    ord_less_set_num: set_num > fun_set_num_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_set_rat: set_rat > fun_set_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_set_real: set_real > fun_set_real_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_less_set_set_nat: ( set_set_nat * set_set_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Uint32__Ouint32_J,type,
    ord_less_set_uint32: ( set_uint32 * set_uint32 ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    ord_le3480810397992357184T_VEBT: set_VEBT_VEBT > fun_set_VEBT_VEBT_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Word__Oword_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_J,type,
    ord_le6726900395242856064l_num1: ( set_wo3913738467083021356l_num1 * set_wo3913738467083021356l_num1 ) > $o ).

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

tff(sy_c_Orderings_Oord__class_Oless_001t__Uint32__Ouint32,type,
    ord_less_uint32: uint32 > fun_uint32_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Word__Oword_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,
    ord_le750835935415966154l_num1: word_N3645301735248828278l_num1 > fun_wo7859709073469921915num1_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Int__Oint_M_Eo_J,type,
    ord_less_fun_int_o: ( fun_int_o * fun_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Nat__Onat_M_Eo_J,type,
    ord_less_fun_nat_o: ( fun_nat_o * fun_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Real__Oreal_M_Eo_J,type,
    ord_less_fun_real_o: ( fun_real_o * fun_real_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    ord_le3534172602983796987VEBT_o: ( fun_VEBT_VEBT_o * fun_VEBT_VEBT_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Assertions__Oassn,type,
    ord_less_eq_assn: fun_assn_fun_assn_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
    ord_le3102999989581377725nteger: fun_Co2081214437782444048eger_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
    ord_le2932123472753598470d_enat: extended_enat > fun_Extended_enat_o ).

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: fun_num_fun_num_o ).

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_le5914376470875661696on_nat: fun_op5794009454290973322_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Num__Onum_J,type,
    ord_le6622620407824499402on_num: fun_op493924057549976310_num_o ).

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le2843612097646854710et_nat: ( option_set_nat * option_set_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: fun_real_fun_real_o ).

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le7084787975880047091nteger: set_Code_integer > fun_se7289222521248229150eger_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_le211207098394363844omplex: set_complex > fun_set_complex_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_eq_set_int: set_int > fun_set_int_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > fun_set_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
    ord_less_eq_set_num: set_num > fun_set_num_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    ord_le171416862856029873n_assn: ( set_Pr5949110396991348497n_assn * set_Pr5949110396991348497n_assn ) > $o ).

tff(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 ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le3146513528884898305at_nat: ( set_Pr1261947904930325089at_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
    ord_le8085105155179020875at_num: ( set_Pr6200539531224447659at_num * set_Pr6200539531224447659at_num ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J_J,type,
    ord_le6438908469242860764T_VEBT: ( set_Pr563407847431865468T_VEBT * set_Pr563407847431865468T_VEBT ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
    ord_le880128212290418581um_num: ( set_Pr8218934625190621173um_num * set_Pr8218934625190621173um_num ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
    ord_less_eq_set_rat: set_rat > fun_set_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_eq_set_real: set_real > fun_set_real_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le6893508408891458716et_nat: ( set_set_nat * set_set_nat ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Uint32__Ouint32_J,type,
    ord_le2219237028632753026uint32: ( set_uint32 * set_uint32 ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > fun_set_VEBT_VEBT_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Word__Oword_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_J,type,
    ord_le5203802739334966412l_num1: ( set_wo3913738467083021356l_num1 * set_wo3913738467083021356l_num1 ) > $o ).

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Uint32__Ouint32,type,
    ord_less_eq_uint32: uint32 > fun_uint32_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Word__Oword_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,
    ord_le3335648743751981014l_num1: word_N3645301735248828278l_num1 > fun_wo7859709073469921915num1_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Int__Oint_Mt__fun_It__Int__Oint_M_Eo_J_J,type,
    ord_le77980682512674322_int_o: ( fun_int_fun_int_o * fun_int_fun_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    ord_le768339454706049738_nat_o: ( fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
    ord_le1526520017675560704_num_o: ( fun_nat_fun_num_o * fun_nat_fun_num_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Nat__Onat_Mt__fun_It__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_M_Eo_J_J,type,
    ord_le2619216865166325287VEBT_o: ( fun_na5540905182366533063VEBT_o * fun_na5540905182366533063VEBT_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Num__Onum_Mt__fun_It__Num__Onum_M_Eo_J_J,type,
    ord_le3306795289226699318_num_o: ( fun_num_fun_num_o * fun_num_fun_num_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
    ord_max_Code_integer: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
    ord_ma741700101516333627d_enat: ( extended_enat * extended_enat ) > extended_enat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
    ord_max_int: fun_int_fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
    ord_max_num: fun_num_fun_num_num ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
    ord_max_rat: ( rat * rat ) > rat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: real > fun_real_real ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_max_set_nat: ( set_nat * set_nat ) > set_nat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Uint32__Ouint32,type,
    ord_max_uint32: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Word__Oword_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,
    ord_ma8239519435860878689l_num1: ( word_N3645301735248828278l_num1 * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
    ord_mi8085742599997312461d_enat: ( extended_enat * extended_enat ) > extended_enat ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
    order_Greatest_nat: fun_nat_o > nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

tff(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J,type,
    partia6726927458685305659Ti_nat: fun_se788439010262289911Heap_o > fun_se3420632608416671075Heap_o ).

tff(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    partia5551857090987368152Ti_nat: fun_se788439010262289911Heap_o > fun_se7475304312357925539Heap_o ).

tff(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J,type,
    partia6039416512482706817Ti_nat: fun_se3101692492007276259on_nat > fun_se1756668854628305023on_nat ).

tff(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    partia2080987842261039902Ti_nat: fun_se3101692492007276259on_nat > fun_se772746884452211909on_nat ).

tff(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J,type,
    partia7782936097874681665Ti_nat: fun_se3996372049210177955_VEBTi > fun_se3949293115670521129_VEBTi ).

tff(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    partia6972460264168101086Ti_nat: fun_se3996372049210177955_VEBTi > fun_se5764602393054361839_VEBTi ).

tff(sy_c_Partial__Function_Ofun__ord_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J,type,
    partia3290229181235258227Ti_nat: fun_He2061078463025973126ap_o_o > fun_fu6850233865993928942ap_o_o ).

tff(sy_c_Partial__Function_Ofun__ord_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
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tff(sy_c_Time__Reasoning_Otime_001_Eo,type,
    time_time_o: ( heap_Time_Heap_o * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Heap__Oarray_I_Eo_J,type,
    time_time_array_o: ( heap_T5660665574680485309rray_o * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Heap__Oarray_It__Int__Oint_J,type,
    time_time_array_int: ( heap_T1346037964561226099ay_int * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Heap__Oarray_It__Nat__Onat_J,type,
    time_time_array_nat: ( heap_T3836121109492952855ay_nat * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_t9122064381910598399_VEBTi: ( heap_T8822477325091257596_VEBTi * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__List__Olist_It__Product____Type__Ounit_J,type,
    time_t4781937132199089312t_unit: ( heap_T7268547540234007069t_unit * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_t3534373299052942712_VEBTi: ( heap_T4980287057938770641_VEBTi * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Nat__Onat,type,
    time_time_nat: ( heap_Time_Heap_nat * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_time_option_nat: ( heap_T2636463487746394924on_nat * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Product____Type__Ounit,type,
    time_t4224138285095624986t_unit: ( heap_T5738788834812785303t_unit * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_time_VEBT_VEBTi: ( heap_T8145700208782473153_VEBTi * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo4422821103128117721l_real: ( filter_real * fun_real_real ) > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
    topolo6980174941875973593q_real: fun_nat_real > $o ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
    topolo2177554685111907308n_real: ( real * set_real ) > filter_real ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
    topolo2815343760600316023s_real: real > filter_real ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
    topolo4055970368930404560y_real: fun_nat_real > $o ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: real > fun_real_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: real > fun_real_real ).

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____1,type,
    type_l31302759751748491nite_1: itself_finite_1 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____2,type,
    type_l31302759751748492nite_2: itself_finite_2 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____3,type,
    type_l31302759751748493nite_3: itself_finite_3 > nat ).

tff(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 ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Onum0,type,
    type_l4264026598287037464l_num0: itself_Numeral_num0 > nat ).

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Onum1,type,
    type_l4264026598287037465l_num1: itself_Numeral_num1 > nat ).

tff(sy_c_Uint32_ORep__uint32_H,type,
    rep_uint32: uint32 > word_N3645301735248828278l_num1 ).

tff(sy_c_Uint32_OUint32,type,
    uint322: fun_Co1554742298434871494uint32 ).

tff(sy_c_Uint32_OUint32__signed,type,
    uint32_signed: fun_Co1554742298434871494uint32 ).

tff(sy_c_Uint32_Odiv0__uint32,type,
    div0_uint32: uint32 > uint32 ).

tff(sy_c_Uint32_Ointeger__of__uint32,type,
    integer_of_uint32: fun_ui1560626348547333958nteger ).

tff(sy_c_Uint32_Ointeger__of__uint32__signed,type,
    intege5370686899274169573signed: uint32 > code_integer ).

tff(sy_c_Uint32_Omod0__uint32,type,
    mod0_uint32: uint32 > uint32 ).

tff(sy_c_Uint32_Oset__bits__aux__uint32,type,
    set_bits_aux_uint32: ( fun_nat_o * nat * uint32 ) > uint32 ).

tff(sy_c_Uint32_Osigned__drop__bit__uint32,type,
    signed489701013188660438uint32: fun_na3857091055356329634uint32 ).

tff(sy_c_Uint32_Ouint32_OAbs__uint32,type,
    abs_uint32: word_N3645301735248828278l_num1 > uint32 ).

tff(sy_c_Uint32_Ouint32_ORep__uint32,type,
    rep_uint322: uint32 > word_N3645301735248828278l_num1 ).

tff(sy_c_Uint32_Ouint32__div,type,
    uint32_div: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Uint32_Ouint32__divmod,type,
    uint32_divmod: ( uint32 * uint32 ) > produc827990862158126777uint32 ).

tff(sy_c_Uint32_Ouint32__mod,type,
    uint32_mod: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Uint32_Ouint32__sdiv,type,
    uint32_sdiv: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Uint32_Ouint32__set__bit,type,
    uint32_set_bit: ( uint32 * code_integer * $o ) > uint32 ).

tff(sy_c_Uint32_Ouint32__shiftl,type,
    uint32_shiftl: uint32 > fun_Co1554742298434871494uint32 ).

tff(sy_c_Uint32_Ouint32__shiftr,type,
    uint32_shiftr: uint32 > fun_Co1554742298434871494uint32 ).

tff(sy_c_Uint32_Ouint32__sshiftr,type,
    uint32_sshiftr: uint32 > fun_Co1554742298434871494uint32 ).

tff(sy_c_Uint32_Ouint32__test__bit,type,
    uint32_test_bit: uint32 > fun_Code_integer_o ).

tff(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: fun_VE597728521808591204at_nat ).

tff(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: fun_VE597728521808591204at_nat ).

tff(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: fun_Pr8877776288780692068_nat_o ).

tff(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: fun_Pr8877776288780692068_nat_o ).

tff(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: fun_VEBT_VEBT_nat ).

tff(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: fun_VE2962686792283380846VEBT_o ).

tff(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: fun_VE597728521808591204at_nat ).

tff(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: fun_VE597728521808591204at_nat ).

tff(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: fun_Pr8877776288780692068_nat_o ).

tff(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: fun_Pr8877776288780692068_nat_o ).

tff(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: fun_VEBT_VEBT_nat ).

tff(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: fun_VE2962686792283380846VEBT_o ).

tff(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: fun_VEBT_VEBT_nat ).

tff(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: fun_VE2962686792283380846VEBT_o ).

tff(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: fun_VE597728521808591204at_nat ).

tff(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: fun_VE597728521808591204at_nat ).

tff(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: fun_Pr8877776288780692068_nat_o ).

tff(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: fun_Pr8877776288780692068_nat_o ).

tff(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: fun_VE597728521808591204at_nat ).

tff(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: fun_VE597728521808591204at_nat ).

tff(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: fun_Pr8877776288780692068_nat_o ).

tff(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: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
    vEBT_V9176841429113362141ildupi: fun_nat_int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
    vEBT_VEBT_Tb: fun_nat_int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
    vEBT_VEBT_Tb2: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
    vEBT_VEBT_highi: fun_na5568033413464243950ap_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
    vEBT_VEBT_lowi: fun_na5568033413464243950ap_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli,type,
    vEBT_VEBT_minNulli: fun_VE7038733415900436564Heap_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli__rel,type,
    vEBT_V5740978063120863272li_rel: fun_VE8636387394336989792EBTi_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001_Eo,type,
    vEBT_V2326993469660664182atei_o: ( nat * heap_Time_Heap_o ) > heap_T844314716496656296list_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Nat__Onat,type,
    vEBT_V7726092123322077554ei_nat: ( nat * heap_Time_Heap_nat ) > heap_T290393402774840812st_nat ).

tff(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 ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Product____Type__Ounit,type,
    vEBT_V7483891112628345579t_unit: ( nat * heap_T5738788834812785303t_unit ) > heap_T7268547540234007069t_unit ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_V1859673955506687831_VEBTi: ( nat * heap_T8145700208782473153_VEBTi ) > heap_T4980287057938770641_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H,type,
    vEBT_V739175172307565963ildupi: fun_na7848093160509028170_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H__rel,type,
    vEBT_V254170901696579886pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__inserti_H,type,
    vEBT_V3964819847710782039nserti: fun_VE2018681615261037117_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__memberi_H,type,
    vEBT_V854960066525838166emberi: fun_VE8181444127144381288Heap_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_OLeafi,type,
    vEBT_Leafi: ( $o * $o ) > vEBT_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_ONodei,type,
    vEBT_Nodei: ( option4927543243414619207at_nat * nat * array_VEBT_VEBTi ) > fun_VE8449866691444886503_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    vEBT_c6104975476656191286Heap_o: ( fun_op1763656958002597022Heap_o * fun_o_1207901889202581752Heap_o ) > fun_VE7038733415900436564Heap_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    vEBT_c1335663792808957512ap_nat: ( fun_op2425996437410981688ap_nat * fun_o_7009304396571837746ap_nat * vEBT_VEBTi ) > heap_Time_Heap_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    vEBT_c6250501799366334488on_nat: ( fun_op5783775701855621896on_nat * fun_o_4049874799390722306on_nat ) > fun_VE2693554162914142864on_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    vEBT_c6028912655521741485_VEBTi: ( fun_op5127434927081926045_VEBTi * fun_o_6972675622308511895_VEBTi ) > fun_VE4765431424076933413_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Nat__Onat,type,
    vEBT_case_VEBTi_nat: ( fun_op7256438278201533260Ti_nat * fun_o_fun_o_nat ) > fun_VEBT_VEBTi_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Osize__VEBTi,type,
    vEBT_size_VEBTi: fun_VEBT_VEBTi_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: fun_VE905575822219544865i_assn ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw__rel,type,
    vEBT_v8524038756793281170aw_rel: fun_Pr3720955412185915546EBTi_o ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
    vEBT_vebt_buildupi: fun_na7848093160509028170_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi__rel,type,
    vEBT_v1230518104690509829pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__inserti,type,
    vEBT_vebt_inserti: fun_VE8118064890490510894_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti,type,
    vEBT_vebt_maxti: fun_VE2693554162914142864on_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti__rel,type,
    vEBT_vebt_maxti_rel: fun_VE8636387394336989792EBTi_o ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__memberi,type,
    vEBT_vebt_memberi: fun_VE7134199479003659425Heap_o ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
    vEBT_vebt_minti: fun_VE2693554162914142864on_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti__rel,type,
    vEBT_vebt_minti_rel: fun_VE8636387394336989792EBTi_o ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( $o * $o ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option4927543243414619207at_nat * nat * list_VEBT_VEBT ) > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Ocase__VEBT_001_Eo,type,
    vEBT_case_VEBT_o: ( fun_op541075054904862521VEBT_o * fun_o_fun_o_o ) > fun_VEBT_VEBT_o ).

tff(sy_c_VEBT__Definitions_OVEBT_Ocase__VEBT_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J_J,type,
    vEBT_c634343235235684882T_VEBT: ( fun_op8229906066235313459T_VEBT * fun_o_8142510167905321106T_VEBT * vEBT_VEBT ) > produc819165548630102716T_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Ois__Node,type,
    vEBT_is_Node: fun_VEBT_VEBT_o ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list_VEBT_VEBT ) > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set_nat ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_o ).

tff(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: fun_VE597728521808591204at_nat ).

tff(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: fun_Pr8877776288780692068_nat_o ).

tff(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: fun_VE597728521808591204at_nat ).

tff(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: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1528199826722428489_VEBTi: ( set_nat * fun_VE905575822219544865i_assn * list_VEBT_VEBT * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6296928887356842470_VEBTi: ( fun_VE905575822219544865i_assn * list_VEBT_VEBT * list_VEBT_VEBTi ) > assn ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: nat > fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: set_nat > fun_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: set_nat > fun_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
    vEBT_V4262088993061758097ft_nat: fun_nat_fun_nat_nat > fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
    vEBT_V819420779217536731ft_num: ( fun_num_fun_num_num * option_num ) > fun_op8989173174992673105on_num ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    vEBT_V613753007643960916it_nat: ( fun_Pr1313076919630728883it_nat * option7339022715339332451it_nat * option7339022715339332451it_nat ) > option7339022715339332451it_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V1502963449132264192at_nat: ( fun_Pr2477239776880934807at_nat * option4927543243414619207at_nat * option4927543243414619207at_nat ) > option4927543243414619207at_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    vEBT_V819568868292977612it_nat: ( fun_Pr3118856740240616547it_nat * option2621746655072343315it_nat * option2621746655072343315it_nat ) > option2621746655072343315it_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > option_nat ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option_nat ).

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: vEBT_VEBT > fun_nat_option_nat ).

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun_Pr8877776288780692068_nat_o ).

tff(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: fun_nat_nat ).

tff(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: fun_nat_nat ).

tff(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: fun_nat_fun_nat_o ).

tff(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: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: fun_VEBT_VEBT_real ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__predi_H,type,
    vEBT_VEBT_vebt_predi: fun_VE3775134097089128360on_nat ).

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__succi_H,type,
    vEBT_VEBT_vebt_succi: fun_VE3775134097089128360on_nat ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__predi,type,
    vEBT_vebt_predi: fun_VE8516060933855021849on_nat ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__succi,type,
    vEBT_vebt_succi: fun_VE8516060933855021849on_nat ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: vEBT_VEBT > fun_nat_option_nat ).

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: fun_nat_fun_nat_o > fun_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: fun_Pr9018117161326028078_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: fun_Pr2028404973800837934_nat_o > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: ( fun_Pr8877776288780692068_nat_o * produc9072475918466114483BT_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    accp_P7675410724331315407_VEBTi: ( fun_Pr3720955412185915546EBTi_o * produc3625547720036274456_VEBTi ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__VEBT____BuildupMemImp__OVEBTi,type,
    accp_VEBT_VEBTi: fun_VE8636387394336989792EBTi_o > fun_VEBT_VEBTi_o ).

tff(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
    accp_VEBT_VEBT: fun_VE2962686792283380846VEBT_o > fun_VEBT_VEBT_o ).

tff(sy_c_Word_Oeven__word_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,
    even_w9054469088133485505l_num1: fun_wo7859709073469921915num1_o ).

tff(sy_c_Word_Osemiring__1__class_Ounsigned_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_001t__Int__Oint,type,
    semiri7338730514057886004m1_int: word_N3645301735248828278l_num1 > int ).

tff(sy_c_Word_Osigned__drop__bit_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,
    signed5000768011106662067l_num1: ( nat * word_N3645301735248828278l_num1 ) > word_N3645301735248828278l_num1 ).

tff(sy_c_aa_001_Eo_001_Eo,type,
    aa_o_o: ( fun_o_o * $o ) > $o ).

tff(sy_c_aa_001_Eo_001t__Assertions__Oassn,type,
    aa_o_assn: ( fun_o_assn * $o ) > assn ).

tff(sy_c_aa_001_Eo_001t__Code____Numeral__Ointeger,type,
    aa_o_Code_integer: ( fun_o_Code_integer * $o ) > code_integer ).

tff(sy_c_aa_001_Eo_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    aa_o_H7399033255965945470Heap_o: ( fun_o_8325599348076409421Heap_o * $o ) > heap_Time_Heap_o ).

tff(sy_c_aa_001_Eo_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    aa_o_H8739796202006397184ap_nat: ( fun_o_4158591058828729991ap_nat * $o ) > heap_Time_Heap_nat ).

tff(sy_c_aa_001_Eo_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    aa_o_H4215342640551218896on_nat: ( fun_o_1929998731553720919on_nat * $o ) > heap_T2636463487746394924on_nat ).

tff(sy_c_aa_001_Eo_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    aa_o_H7577119287286566245_VEBTi: ( fun_o_2238389867833408620_VEBTi * $o ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_aa_001_Eo_001t__Int__Oint,type,
    aa_o_int: ( fun_o_int * $o ) > int ).

tff(sy_c_aa_001_Eo_001t__Nat__Onat,type,
    aa_o_nat: ( fun_o_nat * $o ) > nat ).

tff(sy_c_aa_001_Eo_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
    aa_o_P4714545900465707367T_VEBT: ( fun_o_5254285122974450542T_VEBT * $o ) > produc2504756804600209347T_VEBT ).

tff(sy_c_aa_001_Eo_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_o_P3134438092330336605nt_int: ( fun_o_8988054265482332260nt_int * $o ) > product_prod_int_int ).

tff(sy_c_aa_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_o_P6312935399938478757at_nat: ( fun_o_67843798866261420at_nat * $o ) > product_prod_nat_nat ).

tff(sy_c_aa_001_Eo_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J_J,type,
    aa_o_P8223745691486049304T_VEBT: ( fun_o_3001263932818058983T_VEBT * $o ) > produc819165548630102716T_VEBT ).

tff(sy_c_aa_001_Eo_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    aa_o_P2543913824914997935VEBT_o: ( fun_o_885989297842161846VEBT_o * $o ) > produc334124729049499915VEBT_o ).

tff(sy_c_aa_001_Eo_001t__Real__Oreal,type,
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tff(sy_c_aa_001_Eo_001t__String__Ochar,type,
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tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Complex__Ocomplex_M_Eo_J_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Complex__Ocomplex_J_J,type,
    aa_fun5829985485761010584omplex: ( fun_fu8871020438547428127omplex * fun_VE2006399705683203104omplex ) > fun_fu2816609810865982204omplex ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Int__Oint_M_Eo_J_J_001t__fun_It__Int__Oint_Mt__fun_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
    aa_fun8807180201667926370VEBT_o: ( fun_fu6979804496544055329VEBT_o * fun_VE4235606543851232980_int_o ) > fun_in4578122135805219920VEBT_o ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_fun6249118799011721995at_nat: ( fun_fu5899661017408545610at_nat * fun_VE6480751511521678896_nat_o ) > fun_nat_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__Nat__Onat_Mt__fun_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
    aa_fun2405540735884763746VEBT_o: ( fun_fu1493198444879918881VEBT_o * fun_VE6480751511521678896_nat_o ) > fun_na6887940067898170868VEBT_o ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    aa_fun3621025987637128233BT_nat: ( fun_fu7076988630286034496BT_nat * fun_VE6480751511521678896_nat_o ) > fun_VEBT_VEBT_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_M_Eo_J_J,type,
    aa_fun1998352179508271774_nat_o: ( fun_fu2435797646934184285_nat_o * fun_VE6480751511521678896_nat_o ) > fun_VE6480751511521678896_nat_o ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_J_001t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
    aa_fun3960763739589619457omplex: ( fun_fu6112804207117106256omplex * fun_VE6440932821983644738omplex ) > fun_nat_complex ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    aa_fun2705500642357200114at_nat: ( fun_fu4061198407372677825at_nat * fun_VE597728521808591204at_nat ) > fun_fu5899661017408545610at_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__fun_It__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
    aa_fun1603736613815373464BT_nat: ( fun_fu2828149587716756511BT_nat * fun_VE597728521808591204at_nat ) > fun_fu7076988630286034496BT_nat ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__Nat__Onat_Mt__Real__Oreal_J_J_001t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
    aa_fun6147343916763992189t_real: ( fun_fu830247258886439116t_real * fun_VE2989161857680785728t_real ) > fun_nat_real ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__fun_It__Nat__Onat_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J_J_J_001_Eo,type,
    aa_fun339294776471787884ap_o_o: ( fun_fu2707964056798801139ap_o_o * fun_VE8181444127144381288Heap_o ) > $o ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__fun_It__Nat__Onat_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J_J_J_001_Eo,type,
    aa_fun5369152942363543986_nat_o: ( fun_fu6711510597495717193_nat_o * fun_VE3775134097089128360on_nat ) > $o ).

tff(sy_c_aa_001t__fun_It__VEBT____Definitions__OVEBT_Mt__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__fun_It__Nat__Onat_Mt__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J_J_J_001_Eo,type,
    aa_fun2485637723310249309EBTi_o: ( fun_fu1421233379579251956EBTi_o * fun_VE2018681615261037117_VEBTi ) > $o ).

tff(sy_c_aa_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
    aa_fun3449071308854940378uint32: ( fun_fu713059014528672409uint32 * fun_fu1303640425391265102uint32 ) > fun_fu1303640425391265102uint32 ).

tff(sy_c_aa_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__Uint32__Ouint32_J_J_J_001t__fun_It__fun_It__Nat__Onat_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__Uint32__Ouint32_J_J_J,type,
    aa_fun953596214170897114uint32: ( fun_fu5055813439476669081uint32 * fun_fu5030138910357066062uint32 ) > fun_fu5030138910357066062uint32 ).

tff(sy_c_aa_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_M_Eo_J_J_J_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_M_Eo_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_M_Eo_J_J_J,type,
    aa_fun1804373999164127658eger_o: ( fun_fu8568212769178095465eger_o * fun_fu3466578569608634614eger_o ) > fun_fu3466578569608634614eger_o ).

tff(sy_c_aa_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_J_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Nat__Onat_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Code____Numeral__Ointeger_Mt__fun_I_Eo_Mt__Uint32__Ouint32_J_J_J_J,type,
    aa_fun3432716020469233706uint32: ( fun_fu5407763515825486313uint32 * fun_fu8139443716560175030uint32 ) > fun_fu8139443716560175030uint32 ).

tff(sy_c_aa_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J_001t__fun_It__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_Mt__fun_It__Uint32__Ouint32_Mt__fun_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
    aa_fun3431294132209178672uint32: ( fun_fu6311994054701583983uint32 * fun_fu6749172196426063865uint32 ) > fun_fu6749172196426063865uint32 ).

tff(sy_c_fChoice_001t__Real__Oreal,type,
    fChoice_real: fun_real_o > real ).

tff(sy_c_fequal_001t__Complex__Ocomplex,type,
    fequal_complex: complex > fun_complex_o ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_o ).

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_o ).

tff(sy_c_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    fequal6010348034681699726nt_int: product_prod_int_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_fequal_001t__Real__Oreal,type,
    fequal_real: fun_real_fun_real_o ).

tff(sy_c_fequal_001t__VEBT____Definitions__OVEBT,type,
    fequal_VEBT_VEBT: vEBT_VEBT > fun_VEBT_VEBT_o ).

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

tff(sy_c_member_001t__Code____Numeral__Ointeger,type,
    member_Code_integer: code_integer > fun_se7289222521248229150eger_o ).

tff(sy_c_member_001t__Complex__Ocomplex,type,
    member_complex: complex > fun_set_complex_o ).

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

tff(sy_c_member_001t__List__Olist_I_Eo_J,type,
    member_list_o: ( list_o * set_list_o ) > $o ).

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

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

tff(sy_c_member_001t__List__Olist_It__Real__Oreal_J,type,
    member_list_real: ( list_real * set_list_real ) > $o ).

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

tff(sy_c_member_001t__Num__Onum,type,
    member_num: num > fun_set_num_o ).

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

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

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
    member6310962623043647828_nat_o: ( product_prod_nat_o * set_Pr3149072824959771635_nat_o ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    member4262671552274231302at_int: ( product_prod_nat_int * set_Pr7995236796853374141at_int ) > $o ).

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

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    member9148766508732265716at_num: ( product_prod_nat_num * set_Pr6200539531224447659at_num ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J,type,
    member306291179834725981T_VEBT: ( produc8398139464844984134T_VEBT * set_Pr563407847431865468T_VEBT ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    member557208447399453958t_real: ( produc7716430852924023517t_real * set_Pr320017278500174781t_real ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    member763447850065367567_VEBTi: ( produc214224863196444774_VEBTi * set_Pr1938536134445252166_VEBTi ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
    member8549952807677709168T_VEBT: ( produc8025551001238799321T_VEBT * set_Pr6167073792073659919T_VEBT ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    member7279096912039735102um_num: ( product_prod_num_num * set_Pr8218934625190621173um_num ) > $o ).

tff(sy_c_member_001t__Rat__Orat,type,
    member_rat: rat > fun_set_rat_o ).

tff(sy_c_member_001t__Real__Oreal,type,
    member_real: real > fun_set_real_o ).

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

tff(sy_c_member_001t__VEBT____BuildupMemImp__OVEBTi,type,
    member_VEBT_VEBTi: ( vEBT_VEBTi * set_VEBT_VEBTi ) > $o ).

tff(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
    member_VEBT_VEBT: vEBT_VEBT > fun_set_VEBT_VEBT_o ).

tff(sy_v_f____,type,
    f: fun_VE3775134097089128360on_nat ).

tff(sy_v_ta____,type,
    ta: vEBT_VEBT ).

tff(sy_v_tia____,type,
    tia: vEBT_VEBTi ).

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (11380)
tff(fact_0_refines__case__VEBTi,axiom,
    ! [Tib: vEBT_VEBTi,Ti: vEBT_VEBTi,F1: fun_o_4049874799390722306on_nat,F12: fun_o_4049874799390722306on_nat,F2: fun_op5783775701855621896on_nat,F22: fun_op5783775701855621896on_nat] :
      ( ( Tib = Ti )
     => ( ! [A: $o,B: $o] : refine7594492741263601813on_nat(aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(F1,(A)),(B)),aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(F12,(A)),(B)))
       => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] : refine7594492741263601813on_nat(aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F2,Info),Deg),TreeArray),Summary),aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F22,Info),Deg),TreeArray),Summary))
         => refine7594492741263601813on_nat(aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(F2,F1),Tib),aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(F22,F12),Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_1_refines__case__VEBTi,axiom,
    ! [Tib: vEBT_VEBTi,Ti: vEBT_VEBTi,F1: fun_o_1207901889202581752Heap_o,F12: fun_o_1207901889202581752Heap_o,F2: fun_op1763656958002597022Heap_o,F22: fun_op1763656958002597022Heap_o] :
      ( ( Tib = Ti )
     => ( ! [A: $o,B: $o] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(F1,(A)),(B))),aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(F12,(A)),(B)))
       => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(F2,Info),Deg),TreeArray),Summary)),aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(F22,Info),Deg),TreeArray),Summary))
         => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(F2,F1),Tib)),aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(F22,F12),Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_2_refines__case__VEBTi,axiom,
    ! [Tib: vEBT_VEBTi,Ti: vEBT_VEBTi,F1: fun_o_6972675622308511895_VEBTi,F12: fun_o_6972675622308511895_VEBTi,F2: fun_op5127434927081926045_VEBTi,F22: fun_op5127434927081926045_VEBTi] :
      ( ( Tib = Ti )
     => ( ! [A: $o,B: $o] : refine5565527176597971370_VEBTi(aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(F1,(A)),(B)),aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(F12,(A)),(B)))
       => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] : refine5565527176597971370_VEBTi(aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F2,Info),Deg),TreeArray),Summary),aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F22,Info),Deg),TreeArray),Summary))
         => refine5565527176597971370_VEBTi(aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(F2,F1),Tib),aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(F22,F12),Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_3_greater__shift,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Xb)),aa_nat_option_nat(some_nat,Y)) ) ).

% greater_shift
tff(fact_4_max__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_max_in_set(Xs),Xb)
    <=> ( aa_set_nat_o(member_nat(Xb),Xs)
        & ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xb) ) ) ) ).

% max_in_set_def
tff(fact_5_min__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_min_in_set(Xs),Xb)
    <=> ( aa_set_nat_o(member_nat(Xb),Xs)
        & ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),X) ) ) ) ).

% min_in_set_def
tff(fact_6_power__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,aa_nat_option_nat(some_nat,Xb)),aa_nat_option_nat(some_nat,Y)) = aa_nat_option_nat(some_nat,Z) ) ) ).

% power_shift
tff(fact_7_bit__split__inv,axiom,
    ! [Xb: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),D)),D) = Xb ).

% bit_split_inv
tff(fact_8_high__def,axiom,
    ! [Xb: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% high_def
tff(fact_9_lesseq__shift,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_lesseq,aa_nat_option_nat(some_nat,Xb)),aa_nat_option_nat(some_nat,Y)) ) ).

% lesseq_shift
tff(fact_10_vebt__predi_Osimps,axiom,
    ! [Tb: vEBT_VEBTi,Xb: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_predi,Tb),Xb) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aTP_Lamm_bj(Xb),aTP_Lamm_bk(Xb)),Tb) ).

% vebt_predi.simps
tff(fact_11_VEBT__internal_Ovebt__predi_H_Osimps,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_predi,Tb),Tib),Xb) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_cb(Tb),Xb),aTP_Lamm_bk(Xb)),Tib) ).

% VEBT_internal.vebt_predi'.simps
tff(fact_12_refines__replicate,axiom,
    ! [Fa: heap_T2636463487746394924on_nat,F: heap_T2636463487746394924on_nat,N: nat] :
      ( refine7594492741263601813on_nat(Fa,F)
     => refine1935026298455697829on_nat(vEBT_V792416675989592002on_nat(N,Fa),vEBT_V792416675989592002on_nat(N,F)) ) ).

% refines_replicate
tff(fact_13_refines__replicate,axiom,
    ! [Fa: heap_Time_Heap_o,F: heap_Time_Heap_o,N: nat] :
      ( aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(Fa),F)
     => refine5896690332125372649list_o(vEBT_V2326993469660664182atei_o(N,Fa),vEBT_V2326993469660664182atei_o(N,F)) ) ).

% refines_replicate
tff(fact_14_refines__replicate,axiom,
    ! [Fa: heap_T8145700208782473153_VEBTi,F: heap_T8145700208782473153_VEBTi,N: nat] :
      ( refine5565527176597971370_VEBTi(Fa,F)
     => refine3700189196150522554_VEBTi(vEBT_V1859673955506687831_VEBTi(N,Fa),vEBT_V1859673955506687831_VEBTi(N,F)) ) ).

% refines_replicate
tff(fact_15_power__decreasing__iff,axiom,
    ! [B2: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B2),one_one_Code_integer)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ) ) ).

% power_decreasing_iff
tff(fact_16_power__decreasing__iff,axiom,
    ! [B2: real,M: nat,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ) ) ).

% power_decreasing_iff
tff(fact_17_power__decreasing__iff,axiom,
    ! [B2: rat,M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B2),M)),aa_nat_rat(power_power_rat(B2),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ) ) ).

% power_decreasing_iff
tff(fact_18_power__decreasing__iff,axiom,
    ! [B2: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ) ) ).

% power_decreasing_iff
tff(fact_19_power__decreasing__iff,axiom,
    ! [B2: int,M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),M)),aa_nat_int(power_power_int(B2),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ) ) ).

% power_decreasing_iff
tff(fact_20_zero__less__power2,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A2 != zero_z3403309356797280102nteger ) ) ).

% zero_less_power2
tff(fact_21_zero__less__power2,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A2 != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_22_zero__less__power2,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A2 != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_23_zero__less__power2,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A2 != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_24_power2__less__eq__zero__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)
    <=> ( A2 = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_25_power2__less__eq__zero__iff,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_z3403309356797280102nteger)
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% power2_less_eq_zero_iff
tff(fact_26_power2__less__eq__zero__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)
    <=> ( A2 = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_27_power2__less__eq__zero__iff,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)
    <=> ( A2 = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_28_power2__eq__iff__nonneg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_29_power2__eq__iff__nonneg,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_30_power2__eq__iff__nonneg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_31_power2__eq__iff__nonneg,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_32_power2__eq__iff__nonneg,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_33_bits__1__div__2,axiom,
    divide1791077408188789448l_num1(one_on7727431528512463931l_num1,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = zero_z3563351764282998399l_num1 ).

% bits_1_div_2
tff(fact_34_bits__1__div__2,axiom,
    aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,one_one_uint32),numera9087168376688890119uint32(aa_num_num(bit0,one))) = zero_zero_uint32 ).

% bits_1_div_2
tff(fact_35_bits__1__div__2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% bits_1_div_2
tff(fact_36_bits__1__div__2,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_37_one__div__two__eq__zero,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% one_div_two_eq_zero
tff(fact_38_one__div__two__eq__zero,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_39_power__increasing__iff,axiom,
    ! [B2: code_integer,Xb: nat,Y: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B2),Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_40_power__increasing__iff,axiom,
    ! [B2: real,Xb: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_41_power__increasing__iff,axiom,
    ! [B2: rat,Xb: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B2),Xb)),aa_nat_rat(power_power_rat(B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_42_power__increasing__iff,axiom,
    ! [B2: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_43_power__increasing__iff,axiom,
    ! [B2: int,Xb: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),Xb)),aa_nat_int(power_power_int(B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_44_power__mono__iff,axiom,
    ! [A2: real,B2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_45_power__mono__iff,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N))
          <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_46_power__mono__iff,axiom,
    ! [A2: rat,B2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B2),N))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_47_power__mono__iff,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_48_power__mono__iff,axiom,
    ! [A2: int,B2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(B2),N))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_49_power__strict__decreasing__iff,axiom,
    ! [B2: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B2),one_one_Code_integer)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_50_power__strict__decreasing__iff,axiom,
    ! [B2: real,M: nat,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_51_power__strict__decreasing__iff,axiom,
    ! [B2: rat,M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B2),M)),aa_nat_rat(power_power_rat(B2),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_52_power__strict__decreasing__iff,axiom,
    ! [B2: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_53_power__strict__decreasing__iff,axiom,
    ! [B2: int,M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),M)),aa_nat_int(power_power_int(B2),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_54_zero__eq__power2,axiom,
    ! [A2: rat] :
      ( ( aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_55_zero__eq__power2,axiom,
    ! [A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
    <=> ( A2 = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_56_zero__eq__power2,axiom,
    ! [A2: int] :
      ( ( aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_57_zero__eq__power2,axiom,
    ! [A2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_58_zero__eq__power2,axiom,
    ! [A2: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex )
    <=> ( A2 = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_59_zero__eq__power2,axiom,
    ! [A2: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% zero_eq_power2
tff(fact_60_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% half_negative_int_iff
tff(fact_61_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% half_nonnegative_int_iff
tff(fact_62_bits__div__by__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] : divide1791077408188789448l_num1(A2,zero_z3563351764282998399l_num1) = zero_z3563351764282998399l_num1 ).

% bits_div_by_0
tff(fact_63_bits__div__by__0,axiom,
    ! [A2: uint32] : aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),zero_zero_uint32) = zero_zero_uint32 ).

% bits_div_by_0
tff(fact_64_bits__div__by__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),zero_zero_nat) = zero_zero_nat ).

% bits_div_by_0
tff(fact_65_bits__div__by__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_66_bits__div__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] : divide1791077408188789448l_num1(zero_z3563351764282998399l_num1,A2) = zero_z3563351764282998399l_num1 ).

% bits_div_0
tff(fact_67_bits__div__0,axiom,
    ! [A2: uint32] : aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,zero_zero_uint32),A2) = zero_zero_uint32 ).

% bits_div_0
tff(fact_68_bits__div__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A2) = zero_zero_nat ).

% bits_div_0
tff(fact_69_bits__div__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A2) = zero_zero_int ).

% bits_div_0
tff(fact_70_bits__div__by__1,axiom,
    ! [A2: uint32] : aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),one_one_uint32) = A2 ).

% bits_div_by_1
tff(fact_71_bits__div__by__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),one_one_nat) = A2 ).

% bits_div_by_1
tff(fact_72_bits__div__by__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),one_one_int) = A2 ).

% bits_div_by_1
tff(fact_73_power__one,axiom,
    ! [N: nat] : aa_nat_uint32(power_power_uint32(one_one_uint32),N) = one_one_uint32 ).

% power_one
tff(fact_74_power__one,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(one_one_rat),N) = one_one_rat ).

% power_one
tff(fact_75_power__one,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),N) = one_one_nat ).

% power_one
tff(fact_76_power__one,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(one_one_int),N) = one_one_int ).

% power_one
tff(fact_77_power__one,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),N) = one_one_real ).

% power_one
tff(fact_78_power__one,axiom,
    ! [N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,one_one_complex),N) = one_one_complex ).

% power_one
tff(fact_79_power__one,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),N) = one_one_Code_integer ).

% power_one
tff(fact_80_power__one__right,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),one_one_nat) = A2 ).

% power_one_right
tff(fact_81_power__one__right,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(A2),one_one_nat) = A2 ).

% power_one_right
tff(fact_82_power__one__right,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),one_one_nat) = A2 ).

% power_one_right
tff(fact_83_power__one__right,axiom,
    ! [A2: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),one_one_nat) = A2 ).

% power_one_right
tff(fact_84_power__one__right,axiom,
    ! [A2: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),one_one_nat) = A2 ).

% power_one_right
tff(fact_85_power__inject__exp,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),M) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_86_power__inject__exp,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_87_power__inject__exp,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => ( ( aa_nat_rat(power_power_rat(A2),M) = aa_nat_rat(power_power_rat(A2),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_88_power__inject__exp,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_89_power__inject__exp,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => ( ( aa_nat_int(power_power_int(A2),M) = aa_nat_int(power_power_int(A2),N) )
      <=> ( M = N ) ) ) ).

% power_inject_exp
tff(fact_90_power__zero__numeral,axiom,
    ! [K: num] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(zero_z3563351764282998399l_num1),aa_num_nat(numeral_numeral_nat,K)) = zero_z3563351764282998399l_num1 ).

% power_zero_numeral
tff(fact_91_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_92_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_93_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_int ).

% power_zero_numeral
tff(fact_94_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_95_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_96_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_num_nat(numeral_numeral_nat,K)) = zero_z3403309356797280102nteger ).

% power_zero_numeral
tff(fact_97_div__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat ) ) ).

% div_less
tff(fact_98_nat__zero__less__power__iff,axiom,
    ! [Xb: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_99_power__strict__increasing__iff,axiom,
    ! [B2: code_integer,Xb: nat,Y: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(B2),Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_100_power__strict__increasing__iff,axiom,
    ! [B2: real,Xb: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_101_power__strict__increasing__iff,axiom,
    ! [B2: rat,Xb: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B2),Xb)),aa_nat_rat(power_power_rat(B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_102_power__strict__increasing__iff,axiom,
    ! [B2: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_103_power__strict__increasing__iff,axiom,
    ! [B2: int,Xb: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),Xb)),aa_nat_int(power_power_int(B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_104_power__eq__0__iff,axiom,
    ! [A2: rat,N: nat] :
      ( ( aa_nat_rat(power_power_rat(A2),N) = zero_zero_rat )
    <=> ( ( A2 = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% power_eq_0_iff
tff(fact_105_power__eq__0__iff,axiom,
    ! [A2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% power_eq_0_iff
tff(fact_106_power__eq__0__iff,axiom,
    ! [A2: int,N: nat] :
      ( ( aa_nat_int(power_power_int(A2),N) = zero_zero_int )
    <=> ( ( A2 = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% power_eq_0_iff
tff(fact_107_power__eq__0__iff,axiom,
    ! [A2: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) = zero_zero_real )
    <=> ( ( A2 = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% power_eq_0_iff
tff(fact_108_power__eq__0__iff,axiom,
    ! [A2: complex,N: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N) = zero_zero_complex )
    <=> ( ( A2 = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% power_eq_0_iff
tff(fact_109_power__eq__0__iff,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),N) = zero_z3403309356797280102nteger )
    <=> ( ( A2 = zero_z3403309356797280102nteger )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% power_eq_0_iff
tff(fact_110_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = M )
      <=> ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_111_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_112_not__exp__less__eq__0__int,axiom,
    ! [N: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_113_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: rat,N: nat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A2),N) != zero_zero_rat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_114_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: nat,N: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) != zero_zero_nat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_115_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: int,N: nat] :
      ( ( A2 != zero_zero_int )
     => ( aa_nat_int(power_power_int(A2),N) != zero_zero_int ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_116_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: real,N: nat] :
      ( ( A2 != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) != zero_zero_real ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_117_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: complex,N: nat] :
      ( ( A2 != zero_zero_complex )
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N) != zero_zero_complex ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_118_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: code_integer,N: nat] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),N) != zero_z3403309356797280102nteger ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_119_power__divide,axiom,
    ! [A2: complex,B2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2)),N) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,B2),N)) ).

% power_divide
tff(fact_120_power__divide,axiom,
    ! [A2: real,B2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(A2),B2)),N) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)) ).

% power_divide
tff(fact_121_power__divide,axiom,
    ! [A2: rat,B2: rat,N: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(A2,B2)),N) = divide_divide_rat(aa_nat_rat(power_power_rat(A2),N),aa_nat_rat(power_power_rat(B2),N)) ).

% power_divide
tff(fact_122_div__le__dividend,axiom,
    ! [M: nat,N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M) ).

% div_le_dividend
tff(fact_123_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),K)) ) ).

% div_le_mono
tff(fact_124_zero__le__power,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ) ).

% zero_le_power
tff(fact_125_zero__le__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ) ).

% zero_le_power
tff(fact_126_zero__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),N)) ) ).

% zero_le_power
tff(fact_127_zero__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ) ).

% zero_le_power
tff(fact_128_zero__le__power,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),N)) ) ).

% zero_le_power
tff(fact_129_power__mono,axiom,
    ! [A2: real,B2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)) ) ) ).

% power_mono
tff(fact_130_power__mono,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N)) ) ) ).

% power_mono
tff(fact_131_power__mono,axiom,
    ! [A2: rat,B2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B2),N)) ) ) ).

% power_mono
tff(fact_132_power__mono,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)) ) ) ).

% power_mono
tff(fact_133_power__mono,axiom,
    ! [A2: int,B2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(B2),N)) ) ) ).

% power_mono
tff(fact_134_zero__less__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ) ).

% zero_less_power
tff(fact_135_zero__less__power,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ) ).

% zero_less_power
tff(fact_136_zero__less__power,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),N)) ) ).

% zero_less_power
tff(fact_137_zero__less__power,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ) ).

% zero_less_power
tff(fact_138_zero__less__power,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A2),N)) ) ).

% zero_less_power
tff(fact_139_one__le__power,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ) ).

% one_le_power
tff(fact_140_one__le__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ) ).

% one_le_power
tff(fact_141_one__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A2),N)) ) ).

% one_le_power
tff(fact_142_one__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ) ).

% one_le_power
tff(fact_143_one__le__power,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A2),N)) ) ).

% one_le_power
tff(fact_144_mem__Collect__eq,axiom,
    ! [A2: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),collect_VEBT_VEBT(P))
    <=> aa_VEBT_VEBT_o(P,A2) ) ).

% mem_Collect_eq
tff(fact_145_mem__Collect__eq,axiom,
    ! [A2: real,P: fun_real_o] :
      ( aa_set_real_o(member_real(A2),collect_real(P))
    <=> aa_real_o(P,A2) ) ).

% mem_Collect_eq
tff(fact_146_mem__Collect__eq,axiom,
    ! [A2: nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A2),collect_nat(P))
    <=> aa_nat_o(P,A2) ) ).

% mem_Collect_eq
tff(fact_147_mem__Collect__eq,axiom,
    ! [A2: int,P: fun_int_o] :
      ( aa_set_int_o(member_int(A2),collect_int(P))
    <=> aa_int_o(P,A2) ) ).

% mem_Collect_eq
tff(fact_148_mem__Collect__eq,axiom,
    ! [A2: complex,P: fun_complex_o] :
      ( aa_set_complex_o(member_complex(A2),collect_complex(P))
    <=> aa_complex_o(P,A2) ) ).

% mem_Collect_eq
tff(fact_149_mem__Collect__eq,axiom,
    ! [A2: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      ( member5262025264175285858nt_int(A2,collec213857154873943460nt_int(P))
    <=> aa_Pro4592068669816069595_int_o(P,A2) ) ).

% mem_Collect_eq
tff(fact_150_Collect__mem__eq,axiom,
    ! [A3: set_VEBT_VEBT] : collect_VEBT_VEBT(aTP_Lamm_cc(A3)) = A3 ).

% Collect_mem_eq
tff(fact_151_Collect__mem__eq,axiom,
    ! [A3: set_real] : collect_real(aTP_Lamm_cd(A3)) = A3 ).

% Collect_mem_eq
tff(fact_152_Collect__mem__eq,axiom,
    ! [A3: set_nat] : collect_nat(aTP_Lamm_ce(A3)) = A3 ).

% Collect_mem_eq
tff(fact_153_Collect__mem__eq,axiom,
    ! [A3: set_int] : collect_int(aTP_Lamm_cf(A3)) = A3 ).

% Collect_mem_eq
tff(fact_154_Collect__mem__eq,axiom,
    ! [A3: set_complex] : collect_complex(aTP_Lamm_cg(A3)) = A3 ).

% Collect_mem_eq
tff(fact_155_Collect__mem__eq,axiom,
    ! [A3: set_Pr958786334691620121nt_int] : collec213857154873943460nt_int(aTP_Lamm_ch(A3)) = A3 ).

% Collect_mem_eq
tff(fact_156_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(P,X2)
        <=> aa_nat_o(Q,X2) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_157_Collect__cong,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( aa_int_o(P,X2)
        <=> aa_int_o(Q,X2) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_158_Collect__cong,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ! [X2: complex] :
          ( aa_complex_o(P,X2)
        <=> aa_complex_o(Q,X2) )
     => ( collect_complex(P) = collect_complex(Q) ) ) ).

% Collect_cong
tff(fact_159_Collect__cong,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ! [X2: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(P,X2)
        <=> aa_Pro4592068669816069595_int_o(Q,X2) )
     => ( collec213857154873943460nt_int(P) = collec213857154873943460nt_int(Q) ) ) ).

% Collect_cong
tff(fact_160_power__one__over,axiom,
    ! [A2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),A2)),N) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)) ).

% power_one_over
tff(fact_161_power__one__over,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(one_one_real),A2)),N) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ).

% power_one_over
tff(fact_162_power__one__over,axiom,
    ! [A2: rat,N: nat] : aa_nat_rat(power_power_rat(divide_divide_rat(one_one_rat,A2)),N) = divide_divide_rat(one_one_rat,aa_nat_rat(power_power_rat(A2),N)) ).

% power_one_over
tff(fact_163_power__0,axiom,
    ! [A2: uint32] : aa_nat_uint32(power_power_uint32(A2),zero_zero_nat) = one_one_uint32 ).

% power_0
tff(fact_164_power__0,axiom,
    ! [A2: rat] : aa_nat_rat(power_power_rat(A2),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_165_power__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),zero_zero_nat) = one_one_nat ).

% power_0
tff(fact_166_power__0,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(A2),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_167_power__0,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_168_power__0,axiom,
    ! [A2: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_169_power__0,axiom,
    ! [A2: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),zero_zero_nat) = one_one_Code_integer ).

% power_0
tff(fact_170_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_171_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% nat_power_less_imp_less
tff(fact_172_power__less__imp__less__base,axiom,
    ! [A2: code_integer,N: nat,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B2) ) ) ).

% power_less_imp_less_base
tff(fact_173_power__less__imp__less__base,axiom,
    ! [A2: real,N: nat,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% power_less_imp_less_base
tff(fact_174_power__less__imp__less__base,axiom,
    ! [A2: rat,N: nat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B2),N))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ) ).

% power_less_imp_less_base
tff(fact_175_power__less__imp__less__base,axiom,
    ! [A2: nat,N: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ) ).

% power_less_imp_less_base
tff(fact_176_power__less__imp__less__base,axiom,
    ! [A2: int,N: nat,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(B2),N))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ) ).

% power_less_imp_less_base
tff(fact_177_power__le__one,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),one_one_real) ) ) ).

% power_le_one
tff(fact_178_power__le__one,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),one_one_Code_integer) ) ) ).

% power_le_one
tff(fact_179_power__le__one,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),one_one_rat) ) ) ).

% power_le_one
tff(fact_180_power__le__one,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),one_one_nat) ) ) ).

% power_le_one
tff(fact_181_power__le__one,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),N)),one_one_int) ) ) ).

% power_le_one
tff(fact_182_numeral__Bit0__div__2,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,N) ).

% numeral_Bit0_div_2
tff(fact_183_numeral__Bit0__div__2,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,N) ).

% numeral_Bit0_div_2
tff(fact_184_power__0__left,axiom,
    ! [N: nat] :
      aa_nat_uint32(power_power_uint32(zero_zero_uint32),N) = $ite(N = zero_zero_nat,one_one_uint32,zero_zero_uint32) ).

% power_0_left
tff(fact_185_power__0__left,axiom,
    ! [N: nat] :
      aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(zero_z3563351764282998399l_num1),N) = $ite(N = zero_zero_nat,one_on7727431528512463931l_num1,zero_z3563351764282998399l_num1) ).

% power_0_left
tff(fact_186_power__0__left,axiom,
    ! [N: nat] :
      aa_nat_rat(power_power_rat(zero_zero_rat),N) = $ite(N = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_187_power__0__left,axiom,
    ! [N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = $ite(N = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_188_power__0__left,axiom,
    ! [N: nat] :
      aa_nat_int(power_power_int(zero_zero_int),N) = $ite(N = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_189_power__0__left,axiom,
    ! [N: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = $ite(N = zero_zero_nat,one_one_real,zero_zero_real) ).

% power_0_left
tff(fact_190_power__0__left,axiom,
    ! [N: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),N) = $ite(N = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% power_0_left
tff(fact_191_power__0__left,axiom,
    ! [N: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = $ite(N = zero_zero_nat,one_one_Code_integer,zero_z3403309356797280102nteger) ).

% power_0_left
tff(fact_192_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2)) ) ) ).

% power_strict_increasing
tff(fact_193_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N2)) ) ) ).

% power_strict_increasing
tff(fact_194_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(A2),N2)) ) ) ).

% power_strict_increasing
tff(fact_195_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N2)) ) ) ).

% power_strict_increasing
tff(fact_196_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(A2),N2)) ) ) ).

% power_strict_increasing
tff(fact_197_power__less__imp__less__exp,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% power_less_imp_less_exp
tff(fact_198_power__less__imp__less__exp,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% power_less_imp_less_exp
tff(fact_199_power__less__imp__less__exp,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),M)),aa_nat_rat(power_power_rat(A2),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% power_less_imp_less_exp
tff(fact_200_power__less__imp__less__exp,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% power_less_imp_less_exp
tff(fact_201_power__less__imp__less__exp,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% power_less_imp_less_exp
tff(fact_202_zero__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(zero_z3563351764282998399l_num1),N) = zero_z3563351764282998399l_num1 ) ) ).

% zero_power
tff(fact_203_zero__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),N) = zero_zero_rat ) ) ).

% zero_power
tff(fact_204_zero__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),N) = zero_zero_nat ) ) ).

% zero_power
tff(fact_205_zero__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_int(power_power_int(zero_zero_int),N) = zero_zero_int ) ) ).

% zero_power
tff(fact_206_zero__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),N) = zero_zero_real ) ) ).

% zero_power
tff(fact_207_zero__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),N) = zero_zero_complex ) ) ).

% zero_power
tff(fact_208_zero__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),N) = zero_z3403309356797280102nteger ) ) ).

% zero_power
tff(fact_209_power__increasing,axiom,
    ! [N: nat,N2: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N2)) ) ) ).

% power_increasing
tff(fact_210_power__increasing,axiom,
    ! [N: nat,N2: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2)) ) ) ).

% power_increasing
tff(fact_211_power__increasing,axiom,
    ! [N: nat,N2: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(A2),N2)) ) ) ).

% power_increasing
tff(fact_212_power__increasing,axiom,
    ! [N: nat,N2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N2)) ) ) ).

% power_increasing
tff(fact_213_power__increasing,axiom,
    ! [N: nat,N2: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(A2),N2)) ) ) ).

% power_increasing
tff(fact_214_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% div_greater_zero_iff
tff(fact_215_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M)) ) ) ).

% div_le_mono2
tff(fact_216_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),M) ) ) ).

% div_less_dividend
tff(fact_217_zero__power2,axiom,
    aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(zero_z3563351764282998399l_num1),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_z3563351764282998399l_num1 ).

% zero_power2
tff(fact_218_zero__power2,axiom,
    aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat ).

% zero_power2
tff(fact_219_zero__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% zero_power2
tff(fact_220_zero__power2,axiom,
    aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int ).

% zero_power2
tff(fact_221_zero__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).

% zero_power2
tff(fact_222_zero__power2,axiom,
    aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex ).

% zero_power2
tff(fact_223_zero__power2,axiom,
    aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% zero_power2
tff(fact_224_power__strict__decreasing,axiom,
    ! [N: nat,N2: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_225_power__strict__decreasing,axiom,
    ! [N: nat,N2: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_226_power__strict__decreasing,axiom,
    ! [N: nat,N2: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),N2)),aa_nat_rat(power_power_rat(A2),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_227_power__strict__decreasing,axiom,
    ! [N: nat,N2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_228_power__strict__decreasing,axiom,
    ! [N: nat,N2: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),N2)),aa_nat_int(power_power_int(A2),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_229_power__decreasing,axiom,
    ! [N: nat,N2: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ) ) ) ).

% power_decreasing
tff(fact_230_power__decreasing,axiom,
    ! [N: nat,N2: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N2)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ) ) ) ).

% power_decreasing
tff(fact_231_power__decreasing,axiom,
    ! [N: nat,N2: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N2)),aa_nat_rat(power_power_rat(A2),N)) ) ) ) ).

% power_decreasing
tff(fact_232_power__decreasing,axiom,
    ! [N: nat,N2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ) ) ) ).

% power_decreasing
tff(fact_233_power__decreasing,axiom,
    ! [N: nat,N2: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),N2)),aa_nat_int(power_power_int(A2),N)) ) ) ) ).

% power_decreasing
tff(fact_234_power__eq__imp__eq__base,axiom,
    ! [A2: real,N: nat,B2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_235_power__eq__imp__eq__base,axiom,
    ! [A2: code_integer,N: nat,B2: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),N) = aa_nat_Code_integer(power_8256067586552552935nteger(B2),N) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_236_power__eq__imp__eq__base,axiom,
    ! [A2: rat,N: nat,B2: rat] :
      ( ( aa_nat_rat(power_power_rat(A2),N) = aa_nat_rat(power_power_rat(B2),N) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_237_power__eq__imp__eq__base,axiom,
    ! [A2: nat,N: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_238_power__eq__imp__eq__base,axiom,
    ! [A2: int,N: nat,B2: int] :
      ( ( aa_nat_int(power_power_int(A2),N) = aa_nat_int(power_power_int(B2),N) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_239_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: real,B2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_240_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: code_integer,B2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
         => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),N) = aa_nat_Code_integer(power_8256067586552552935nteger(B2),N) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_241_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: rat,B2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
         => ( ( aa_nat_rat(power_power_rat(A2),N) = aa_nat_rat(power_power_rat(B2),N) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_242_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_243_power__eq__iff__eq__base,axiom,
    ! [N: nat,A2: int,B2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
         => ( ( aa_nat_int(power_power_int(A2),N) = aa_nat_int(power_power_int(B2),N) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_244_one__power2,axiom,
    aa_nat_uint32(power_power_uint32(one_one_uint32),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_uint32 ).

% one_power2
tff(fact_245_one__power2,axiom,
    aa_nat_rat(power_power_rat(one_one_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat ).

% one_power2
tff(fact_246_one__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_power2
tff(fact_247_one__power2,axiom,
    aa_nat_int(power_power_int(one_one_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int ).

% one_power2
tff(fact_248_one__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real ).

% one_power2
tff(fact_249_one__power2,axiom,
    aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,one_one_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex ).

% one_power2
tff(fact_250_one__power2,axiom,
    aa_nat_Code_integer(power_8256067586552552935nteger(one_one_Code_integer),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_power2
tff(fact_251_power__le__imp__le__exp,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_le_imp_le_exp
tff(fact_252_power__le__imp__le__exp,axiom,
    ! [A2: real,M: nat,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_le_imp_le_exp
tff(fact_253_power__le__imp__le__exp,axiom,
    ! [A2: rat,M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),M)),aa_nat_rat(power_power_rat(A2),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_le_imp_le_exp
tff(fact_254_power__le__imp__le__exp,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_le_imp_le_exp
tff(fact_255_power__le__imp__le__exp,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_le_imp_le_exp
tff(fact_256_self__le__power,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ) ) ).

% self_le_power
tff(fact_257_self__le__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ) ) ).

% self_le_power
tff(fact_258_self__le__power,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_nat_rat(power_power_rat(A2),N)) ) ) ).

% self_le_power
tff(fact_259_self__le__power,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ) ) ).

% self_le_power
tff(fact_260_self__le__power,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(A2),N)) ) ) ).

% self_le_power
tff(fact_261_one__less__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ) ) ).

% one_less_power
tff(fact_262_one__less__power,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ) ) ).

% one_less_power
tff(fact_263_one__less__power,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A2),N)) ) ) ).

% one_less_power
tff(fact_264_one__less__power,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ) ) ).

% one_less_power
tff(fact_265_one__less__power,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A2),N)) ) ) ).

% one_less_power
tff(fact_266_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% power2_nat_le_imp_le
tff(fact_267_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% power2_nat_le_eq_le
tff(fact_268_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)) ) ).

% self_le_ge2_pow
tff(fact_269_power2__le__imp__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_270_power2__le__imp__le,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_271_power2__le__imp__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_272_power2__le__imp__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_273_power2__le__imp__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_274_power2__eq__imp__eq,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_275_power2__eq__imp__eq,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_276_power2__eq__imp__eq,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_277_power2__eq__imp__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_278_power2__eq__imp__eq,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_279_zero__le__power2,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_280_zero__le__power2,axiom,
    ! [A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_281_zero__le__power2,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_282_zero__le__power2,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_283_power2__less__0,axiom,
    ! [A2: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_z3403309356797280102nteger) ).

% power2_less_0
tff(fact_284_power2__less__0,axiom,
    ! [A2: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real) ).

% power2_less_0
tff(fact_285_power2__less__0,axiom,
    ! [A2: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat) ).

% power2_less_0
tff(fact_286_power2__less__0,axiom,
    ! [A2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int) ).

% power2_less_0
tff(fact_287_power__strict__mono,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N)) ) ) ) ).

% power_strict_mono
tff(fact_288_power__strict__mono,axiom,
    ! [A2: real,B2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)) ) ) ) ).

% power_strict_mono
tff(fact_289_power__strict__mono,axiom,
    ! [A2: rat,B2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B2),N)) ) ) ) ).

% power_strict_mono
tff(fact_290_power__strict__mono,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)) ) ) ) ).

% power_strict_mono
tff(fact_291_power__strict__mono,axiom,
    ! [A2: int,B2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(B2),N)) ) ) ) ).

% power_strict_mono
tff(fact_292_power2__less__imp__less,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_293_power2__less__imp__less,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_294_power2__less__imp__less,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_295_power2__less__imp__less,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_296_power2__less__imp__less,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_297_le__divide__eq__1__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B2),A2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ).

% le_divide_eq_1_pos
tff(fact_298_le__divide__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),divide_divide_rat(B2,A2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ) ).

% le_divide_eq_1_pos
tff(fact_299_le__divide__eq__1__neg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B2),A2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ) ).

% le_divide_eq_1_neg
tff(fact_300_le__divide__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),divide_divide_rat(B2,A2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ) ).

% le_divide_eq_1_neg
tff(fact_301_divide__le__eq__1__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B2),A2)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ) ).

% divide_le_eq_1_pos
tff(fact_302_divide__le__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B2,A2)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ) ).

% divide_le_eq_1_pos
tff(fact_303_divide__le__eq__1__neg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B2),A2)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ).

% divide_le_eq_1_neg
tff(fact_304_divide__le__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B2,A2)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ) ).

% divide_le_eq_1_neg
tff(fact_305_zero__less__divide__1__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(one_one_real),A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2) ) ).

% zero_less_divide_1_iff
tff(fact_306_zero__less__divide__1__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(one_one_rat,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ).

% zero_less_divide_1_iff
tff(fact_307_less__divide__eq__1__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B2),A2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% less_divide_eq_1_pos
tff(fact_308_less__divide__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),divide_divide_rat(B2,A2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ) ).

% less_divide_eq_1_pos
tff(fact_309_less__divide__eq__1__neg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B2),A2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ) ).

% less_divide_eq_1_neg
tff(fact_310_less__divide__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),divide_divide_rat(B2,A2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ) ).

% less_divide_eq_1_neg
tff(fact_311_divide__less__eq__1__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B2),A2)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ) ).

% divide_less_eq_1_pos
tff(fact_312_divide__less__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B2,A2)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ) ).

% divide_less_eq_1_pos
tff(fact_313_divide__less__eq__1__neg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B2),A2)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% divide_less_eq_1_neg
tff(fact_314_divide__less__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B2,A2)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ) ).

% divide_less_eq_1_neg
tff(fact_315_divide__less__0__1__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(one_one_real),A2)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real) ) ).

% divide_less_0_1_iff
tff(fact_316_divide__less__0__1__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(one_one_rat,A2)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_317_zero__le__divide__1__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(one_one_real),A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ).

% zero_le_divide_1_iff
tff(fact_318_zero__le__divide__1__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(one_one_rat,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ).

% zero_le_divide_1_iff
tff(fact_319_divide__le__0__1__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),A2)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) ) ).

% divide_le_0_1_iff
tff(fact_320_divide__le__0__1__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(one_one_rat,A2)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% divide_le_0_1_iff
tff(fact_321_divide__eq__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(divide_divide_real(A2),B2) = zero_zero_real )
    <=> ( ( A2 = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_322_divide__eq__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( divide_divide_rat(A2,B2) = zero_zero_rat )
    <=> ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_323_divide__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( aa_real_real(divide_divide_real(C),A2) = aa_real_real(divide_divide_real(C),B2) )
    <=> ( ( C = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_left
tff(fact_324_divide__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( divide_divide_rat(C,A2) = divide_divide_rat(C,B2) )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_left
tff(fact_325_divide__cancel__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( aa_real_real(divide_divide_real(A2),C) = aa_real_real(divide_divide_real(B2),C) )
    <=> ( ( C = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_right
tff(fact_326_divide__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( divide_divide_rat(A2,C) = divide_divide_rat(B2,C) )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_right
tff(fact_327_division__ring__divide__zero,axiom,
    ! [A2: real] : aa_real_real(divide_divide_real(A2),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_328_division__ring__divide__zero,axiom,
    ! [A2: rat] : divide_divide_rat(A2,zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_329_divide__eq__1__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(divide_divide_real(A2),B2) = one_one_real )
    <=> ( ( B2 != zero_zero_real )
        & ( A2 = B2 ) ) ) ).

% divide_eq_1_iff
tff(fact_330_divide__eq__1__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( divide_divide_rat(A2,B2) = one_one_rat )
    <=> ( ( B2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% divide_eq_1_iff
tff(fact_331_one__eq__divide__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( one_one_real = aa_real_real(divide_divide_real(A2),B2) )
    <=> ( ( B2 != zero_zero_real )
        & ( A2 = B2 ) ) ) ).

% one_eq_divide_iff
tff(fact_332_one__eq__divide__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( one_one_rat = divide_divide_rat(A2,B2) )
    <=> ( ( B2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% one_eq_divide_iff
tff(fact_333_divide__self,axiom,
    ! [A2: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A2),A2) = one_one_real ) ) ).

% divide_self
tff(fact_334_divide__self,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( divide_divide_rat(A2,A2) = one_one_rat ) ) ).

% divide_self
tff(fact_335_divide__self__if,axiom,
    ! [A2: real] :
      aa_real_real(divide_divide_real(A2),A2) = $ite(A2 = zero_zero_real,zero_zero_real,one_one_real) ).

% divide_self_if
tff(fact_336_divide__self__if,axiom,
    ! [A2: rat] :
      divide_divide_rat(A2,A2) = $ite(A2 = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_337_divide__eq__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( ( aa_real_real(divide_divide_real(B2),A2) = one_one_real )
    <=> ( ( A2 != zero_zero_real )
        & ( A2 = B2 ) ) ) ).

% divide_eq_eq_1
tff(fact_338_divide__eq__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( divide_divide_rat(B2,A2) = one_one_rat )
    <=> ( ( A2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% divide_eq_eq_1
tff(fact_339_eq__divide__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( ( one_one_real = aa_real_real(divide_divide_real(B2),A2) )
    <=> ( ( A2 != zero_zero_real )
        & ( A2 = B2 ) ) ) ).

% eq_divide_eq_1
tff(fact_340_eq__divide__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( one_one_rat = divide_divide_rat(B2,A2) )
    <=> ( ( A2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% eq_divide_eq_1
tff(fact_341_one__divide__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( aa_real_real(divide_divide_real(one_one_real),A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_342_one__divide__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( divide_divide_rat(one_one_rat,A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_343_zero__eq__1__divide__iff,axiom,
    ! [A2: real] :
      ( ( zero_zero_real = aa_real_real(divide_divide_real(one_one_real),A2) )
    <=> ( A2 = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_344_zero__eq__1__divide__iff,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat = divide_divide_rat(one_one_rat,A2) )
    <=> ( A2 = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_345_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_346_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_347_linordered__field__no__lb,axiom,
    ! [X3: real] :
    ? [Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X3) ).

% linordered_field_no_lb
tff(fact_348_linordered__field__no__lb,axiom,
    ! [X3: rat] :
    ? [Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X3) ).

% linordered_field_no_lb
tff(fact_349_linordered__field__no__ub,axiom,
    ! [X3: real] :
    ? [X_1: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X3),X_1) ).

% linordered_field_no_ub
tff(fact_350_linordered__field__no__ub,axiom,
    ! [X3: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),X_1) ).

% linordered_field_no_ub
tff(fact_351_divide__le__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A2),B2)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) ) ) ).

% divide_le_0_iff
tff(fact_352_divide__le__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A2,B2)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) ) ) ).

% divide_le_0_iff
tff(fact_353_divide__right__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A2),C)),aa_real_real(divide_divide_real(B2),C)) ) ) ).

% divide_right_mono
tff(fact_354_divide__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A2,C)),divide_divide_rat(B2,C)) ) ) ).

% divide_right_mono
tff(fact_355_zero__le__divide__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(A2),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) ) ) ) ).

% zero_le_divide_iff
tff(fact_356_zero__le__divide__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(A2,B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_357_divide__nonneg__nonneg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_358_divide__nonneg__nonneg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(Xb,Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_359_divide__nonneg__nonpos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_360_divide__nonneg__nonpos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_361_divide__nonpos__nonneg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_362_divide__nonpos__nonneg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_363_divide__nonpos__nonpos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_364_divide__nonpos__nonpos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(Xb,Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_365_divide__right__mono__neg,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B2),C)),aa_real_real(divide_divide_real(A2),C)) ) ) ).

% divide_right_mono_neg
tff(fact_366_divide__right__mono__neg,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B2,C)),divide_divide_rat(A2,C)) ) ) ).

% divide_right_mono_neg
tff(fact_367_divide__neg__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Y)) ) ) ).

% divide_neg_neg
tff(fact_368_divide__neg__neg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(Xb,Y)) ) ) ).

% divide_neg_neg
tff(fact_369_divide__neg__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xb),Y)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_370_divide__neg__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_371_divide__pos__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xb),Y)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_372_divide__pos__neg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_373_divide__pos__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Y)) ) ) ).

% divide_pos_pos
tff(fact_374_divide__pos__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(Xb,Y)) ) ) ).

% divide_pos_pos
tff(fact_375_divide__less__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(A2),B2)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ) ) ).

% divide_less_0_iff
tff(fact_376_divide__less__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A2,B2)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ) ) ).

% divide_less_0_iff
tff(fact_377_divide__less__cancel,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(A2),C)),aa_real_real(divide_divide_real(B2),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_378_divide__less__cancel,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A2,C)),divide_divide_rat(B2,C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_379_zero__less__divide__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A2),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real) ) ) ) ).

% zero_less_divide_iff
tff(fact_380_zero__less__divide__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(A2,B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_381_divide__strict__right__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(A2),C)),aa_real_real(divide_divide_real(B2),C)) ) ) ).

% divide_strict_right_mono
tff(fact_382_divide__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A2,C)),divide_divide_rat(B2,C)) ) ) ).

% divide_strict_right_mono
tff(fact_383_divide__strict__right__mono__neg,axiom,
    ! [B2: real,A2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(A2),C)),aa_real_real(divide_divide_real(B2),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_384_divide__strict__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(A2,C)),divide_divide_rat(B2,C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_385_right__inverse__eq,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( aa_real_real(divide_divide_real(A2),B2) = one_one_real )
      <=> ( A2 = B2 ) ) ) ).

% right_inverse_eq
tff(fact_386_right__inverse__eq,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( divide_divide_rat(A2,B2) = one_one_rat )
      <=> ( A2 = B2 ) ) ) ).

% right_inverse_eq
tff(fact_387_frac__le,axiom,
    ! [Y: real,Xb: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Z)),aa_real_real(divide_divide_real(Y),W)) ) ) ) ) ).

% frac_le
tff(fact_388_frac__le,axiom,
    ! [Y: rat,Xb: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xb,Z)),divide_divide_rat(Y,W)) ) ) ) ) ).

% frac_le
tff(fact_389_frac__less,axiom,
    ! [Xb: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xb),Z)),aa_real_real(divide_divide_real(Y),W)) ) ) ) ) ).

% frac_less
tff(fact_390_frac__less,axiom,
    ! [Xb: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xb,Z)),divide_divide_rat(Y,W)) ) ) ) ) ).

% frac_less
tff(fact_391_frac__less2,axiom,
    ! [Xb: real,Y: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xb),Z)),aa_real_real(divide_divide_real(Y),W)) ) ) ) ) ).

% frac_less2
tff(fact_392_frac__less2,axiom,
    ! [Xb: rat,Y: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xb,Z)),divide_divide_rat(Y,W)) ) ) ) ) ).

% frac_less2
tff(fact_393_divide__le__cancel,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A2),C)),aa_real_real(divide_divide_real(B2),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ) ) ).

% divide_le_cancel
tff(fact_394_divide__le__cancel,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A2,C)),divide_divide_rat(B2,C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ) ) ).

% divide_le_cancel
tff(fact_395_divide__nonneg__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_396_divide__nonneg__neg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_397_divide__nonneg__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_398_divide__nonneg__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(Xb,Y)) ) ) ).

% divide_nonneg_pos
tff(fact_399_divide__nonpos__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_400_divide__nonpos__neg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(Xb,Y)) ) ) ).

% divide_nonpos_neg
tff(fact_401_divide__nonpos__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_402_divide__nonpos__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xb,Y)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_403_divide__less__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B2),A2)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) )
        | ( A2 = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_404_divide__less__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B2,A2)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) )
        | ( A2 = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_405_less__divide__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B2),A2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ) ) ).

% less_divide_eq_1
tff(fact_406_less__divide__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),divide_divide_rat(B2,A2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ) ) ).

% less_divide_eq_1
tff(fact_407_divide__le__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B2),A2)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) )
        | ( A2 = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_408_divide__le__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B2,A2)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) )
        | ( A2 = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_409_le__divide__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B2),A2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ) ) ).

% le_divide_eq_1
tff(fact_410_le__divide__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),divide_divide_rat(B2,A2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ) ) ).

% le_divide_eq_1
tff(fact_411_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),numeral_numeral_real(N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),N) ) ).

% one_less_numeral_iff
tff(fact_412_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),numeral_numeral_rat(N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),N) ) ).

% one_less_numeral_iff
tff(fact_413_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),N) ) ).

% one_less_numeral_iff
tff(fact_414_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),N) ) ).

% one_less_numeral_iff
tff(fact_415_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(N)),one_one_real)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N),one) ) ).

% numeral_le_one_iff
tff(fact_416_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(N)),one_one_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N),one) ) ).

% numeral_le_one_iff
tff(fact_417_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N),one) ) ).

% numeral_le_one_iff
tff(fact_418_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),one_one_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N),one) ) ).

% numeral_le_one_iff
tff(fact_419_int__div__same__is__1,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = A2 )
      <=> ( B2 = one_one_int ) ) ) ).

% int_div_same_is_1
tff(fact_420_option_Ocollapse,axiom,
    ! [Option: option2621746655072343315it_nat] :
      ( ( Option != none_P1551326421579882414it_nat )
     => ( some_P2407035485129114418it_nat(the_Pr3501439614016493281it_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_421_option_Ocollapse,axiom,
    ! [Option: option7339022715339332451it_nat] :
      ( ( Option != none_P7668321371905463026it_nat )
     => ( some_P468703482102919278it_nat(the_Pr5838048819577852031it_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_422_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( aa_nat_option_nat(some_nat,the_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_423_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_424_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) = Option ) ) ).

% option.collapse
tff(fact_425_prod_Ocollapse,axiom,
    ! [Prod: product_prod_num_num] : aa_num3156681643174609981um_num(product_Pair_num_num(aa_Pro759708478137009011um_num(product_fst_num_num,Prod)),aa_Pro759708478137009011um_num(product_snd_num_num,Prod)) = Prod ).

% prod.collapse
tff(fact_426_prod_Ocollapse,axiom,
    ! [Prod: produc8398139464844984134T_VEBT] : aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(aa_Pro6878622105917549268BT_nat(produc758997459209783180T_VEBT,Prod)),aa_Pro3664079603483438877T_VEBT(produc2084898568784432842T_VEBT,Prod)) = Prod ).

% prod.collapse
tff(fact_427_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_num] : aa_num5026351239867140595at_num(product_Pair_nat_num(aa_Pro6261568864010571487um_nat(product_fst_nat_num,Prod)),aa_Pro2818900650166350121um_num(product_snd_nat_num,Prod)) = Prod ).

% prod.collapse
tff(fact_428_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) = Prod ).

% prod.collapse
tff(fact_429_prod_Ocollapse,axiom,
    ! [Prod: product_prod_int_int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) = Prod ).

% prod.collapse
tff(fact_430_prod_Ocollapse,axiom,
    ! [Prod: produc6575502325842934193n_assn] : aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Prod)),aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Prod)) = Prod ).

% prod.collapse
tff(fact_431_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_432_less__one,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),one_one_nat)
    <=> ( N = zero_zero_nat ) ) ).

% less_one
tff(fact_433_div__self,axiom,
    ! [A2: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A2),A2) = one_one_real ) ) ).

% div_self
tff(fact_434_div__self,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( divide_divide_rat(A2,A2) = one_one_rat ) ) ).

% div_self
tff(fact_435_div__self,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),A2) = one_one_nat ) ) ).

% div_self
tff(fact_436_div__self,axiom,
    ! [A2: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),A2) = one_one_int ) ) ).

% div_self
tff(fact_437_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real(N) = one_one_real )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_438_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat(N) = one_one_rat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_439_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_nat(numeral_numeral_nat,N) = one_one_nat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_440_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( aa_num_int(numeral_numeral_int,N) = one_one_int )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_441_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real = numeral_numeral_real(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_442_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat = numeral_numeral_rat(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_443_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_444_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_445_two__pow__div__gt__le,axiom,
    ! [V: nat,N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,V),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% two_pow_div_gt_le
tff(fact_446_vebt__succi_Osimps,axiom,
    ! [Tb: vEBT_VEBTi,Xb: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_succi,Tb),Xb) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aTP_Lamm_cp(Xb),aTP_Lamm_cq(Xb)),Tb) ).

% vebt_succi.simps
tff(fact_447_less__shift,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,Xb)),aa_nat_option_nat(some_nat,Y)) ) ).

% less_shift
tff(fact_448_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_real(M) = numeral_numeral_real(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_449_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_rat(M) = numeral_numeral_rat(N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_450_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_nat(numeral_numeral_nat,M) = aa_num_nat(numeral_numeral_nat,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_451_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_int(numeral_numeral_int,M) = aa_num_int(numeral_numeral_int,N) )
    <=> ( M = N ) ) ).

% numeral_eq_iff
tff(fact_452_old_Oprod_Oinject,axiom,
    ! [A2: num,B2: num,A4: num,B3: num] :
      ( ( aa_num3156681643174609981um_num(product_Pair_num_num(A2),B2) = aa_num3156681643174609981um_num(product_Pair_num_num(A4),B3) )
    <=> ( ( A2 = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
tff(fact_453_old_Oprod_Oinject,axiom,
    ! [A2: nat,B2: produc4813437837504472865T_VEBT,A4: nat,B3: produc4813437837504472865T_VEBT] :
      ( ( aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A2),B2) = aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A4),B3) )
    <=> ( ( A2 = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
tff(fact_454_old_Oprod_Oinject,axiom,
    ! [A2: nat,B2: num,A4: nat,B3: num] :
      ( ( aa_num5026351239867140595at_num(product_Pair_nat_num(A2),B2) = aa_num5026351239867140595at_num(product_Pair_nat_num(A4),B3) )
    <=> ( ( A2 = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
tff(fact_455_old_Oprod_Oinject,axiom,
    ! [A2: nat,B2: nat,A4: nat,B3: nat] :
      ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A4),B3) )
    <=> ( ( A2 = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
tff(fact_456_old_Oprod_Oinject,axiom,
    ! [A2: int,B2: int,A4: int,B3: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B3) )
    <=> ( ( A2 = A4 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
tff(fact_457_prod_Oinject,axiom,
    ! [X1: num,X22: num,Y1: num,Y22: num] :
      ( ( aa_num3156681643174609981um_num(product_Pair_num_num(X1),X22) = aa_num3156681643174609981um_num(product_Pair_num_num(Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_458_prod_Oinject,axiom,
    ! [X1: nat,X22: produc4813437837504472865T_VEBT,Y1: nat,Y22: produc4813437837504472865T_VEBT] :
      ( ( aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(X1),X22) = aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_459_prod_Oinject,axiom,
    ! [X1: nat,X22: num,Y1: nat,Y22: num] :
      ( ( aa_num5026351239867140595at_num(product_Pair_nat_num(X1),X22) = aa_num5026351239867140595at_num(product_Pair_nat_num(Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_460_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
      ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_461_prod_Oinject,axiom,
    ! [X1: int,X22: int,Y1: int,Y22: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_462_option_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( aa_nat_option_nat(some_nat,X22) = aa_nat_option_nat(some_nat,Y22) )
    <=> ( X22 = Y22 ) ) ).

% option.inject
tff(fact_463_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
tff(fact_464_option_Oinject,axiom,
    ! [X22: num,Y22: num] :
      ( ( aa_num_option_num(some_num,X22) = aa_num_option_num(some_num,Y22) )
    <=> ( X22 = Y22 ) ) ).

% option.inject
tff(fact_465_div__by__0,axiom,
    ! [A2: real] : aa_real_real(divide_divide_real(A2),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_466_div__by__0,axiom,
    ! [A2: rat] : divide_divide_rat(A2,zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_467_div__by__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),zero_zero_nat) = zero_zero_nat ).

% div_by_0
tff(fact_468_div__by__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_469_div__0,axiom,
    ! [A2: real] : aa_real_real(divide_divide_real(zero_zero_real),A2) = zero_zero_real ).

% div_0
tff(fact_470_div__0,axiom,
    ! [A2: rat] : divide_divide_rat(zero_zero_rat,A2) = zero_zero_rat ).

% div_0
tff(fact_471_div__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A2) = zero_zero_nat ).

% div_0
tff(fact_472_div__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A2) = zero_zero_int ).

% div_0
tff(fact_473_div__by__1,axiom,
    ! [A2: real] : aa_real_real(divide_divide_real(A2),one_one_real) = A2 ).

% div_by_1
tff(fact_474_div__by__1,axiom,
    ! [A2: rat] : divide_divide_rat(A2,one_one_rat) = A2 ).

% div_by_1
tff(fact_475_div__by__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),one_one_nat) = A2 ).

% div_by_1
tff(fact_476_div__by__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),one_one_int) = A2 ).

% div_by_1
tff(fact_477_less__nat__zero__code,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_478_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% neq0_conv
tff(fact_479_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_480_le0,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),N) ).

% le0
tff(fact_481_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2) ).

% bot_nat_0.extremum
tff(fact_482_not__Some__eq,axiom,
    ! [Xb: option2621746655072343315it_nat] :
      ( ! [Y3: produc120671012495760973it_nat] : Xb != some_P2407035485129114418it_nat(Y3)
    <=> ( Xb = none_P1551326421579882414it_nat ) ) ).

% not_Some_eq
tff(fact_483_not__Some__eq,axiom,
    ! [Xb: option7339022715339332451it_nat] :
      ( ! [Y3: produc8047831477865546771it_nat] : Xb != some_P468703482102919278it_nat(Y3)
    <=> ( Xb = none_P7668321371905463026it_nat ) ) ).

% not_Some_eq
tff(fact_484_not__Some__eq,axiom,
    ! [Xb: option_nat] :
      ( ! [Y3: nat] : Xb != aa_nat_option_nat(some_nat,Y3)
    <=> ( Xb = none_nat ) ) ).

% not_Some_eq
tff(fact_485_not__Some__eq,axiom,
    ! [Xb: option4927543243414619207at_nat] :
      ( ! [Y3: product_prod_nat_nat] : Xb != some_P7363390416028606310at_nat(Y3)
    <=> ( Xb = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_486_not__Some__eq,axiom,
    ! [Xb: option_num] :
      ( ! [Y3: num] : Xb != aa_num_option_num(some_num,Y3)
    <=> ( Xb = none_num ) ) ).

% not_Some_eq
tff(fact_487_not__None__eq,axiom,
    ! [Xb: option2621746655072343315it_nat] :
      ( ( Xb != none_P1551326421579882414it_nat )
    <=> ? [Y3: produc120671012495760973it_nat] : Xb = some_P2407035485129114418it_nat(Y3) ) ).

% not_None_eq
tff(fact_488_not__None__eq,axiom,
    ! [Xb: option7339022715339332451it_nat] :
      ( ( Xb != none_P7668321371905463026it_nat )
    <=> ? [Y3: produc8047831477865546771it_nat] : Xb = some_P468703482102919278it_nat(Y3) ) ).

% not_None_eq
tff(fact_489_not__None__eq,axiom,
    ! [Xb: option_nat] :
      ( ( Xb != none_nat )
    <=> ? [Y3: nat] : Xb = aa_nat_option_nat(some_nat,Y3) ) ).

% not_None_eq
tff(fact_490_not__None__eq,axiom,
    ! [Xb: option4927543243414619207at_nat] :
      ( ( Xb != none_P5556105721700978146at_nat )
    <=> ? [Y3: product_prod_nat_nat] : Xb = some_P7363390416028606310at_nat(Y3) ) ).

% not_None_eq
tff(fact_491_not__None__eq,axiom,
    ! [Xb: option_num] :
      ( ( Xb != none_num )
    <=> ? [Y3: num] : Xb = aa_num_option_num(some_num,Y3) ) ).

% not_None_eq
tff(fact_492_case__prod__conv,axiom,
    ! [Fa: fun_na4138050610385705875at_nat,A2: nat,B2: nat] : aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(Fa),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(Fa,A2),B2) ).

% case_prod_conv
tff(fact_493_case__prod__conv,axiom,
    ! [Fa: fun_nat_fun_nat_o,A2: nat,B2: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(Fa),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(Fa,A2),B2) ) ).

% case_prod_conv
tff(fact_494_case__prod__conv,axiom,
    ! [Fa: fun_in1696471567231710611nt_int,A2: int,B2: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Fa,A2),B2) ).

% case_prod_conv
tff(fact_495_case__prod__conv,axiom,
    ! [Fa: fun_int_fun_int_o,A2: int,B2: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(Fa,A2),B2) ) ).

% case_prod_conv
tff(fact_496_case__prod__conv,axiom,
    ! [Fa: fun_int_fun_int_int,A2: int,B2: int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(Fa,A2),B2) ).

% case_prod_conv
tff(fact_497_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(M)),numeral_numeral_real(N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N) ) ).

% numeral_le_iff
tff(fact_498_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(M)),numeral_numeral_rat(N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N) ) ).

% numeral_le_iff
tff(fact_499_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N) ) ).

% numeral_le_iff
tff(fact_500_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N) ) ).

% numeral_le_iff
tff(fact_501_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(M)),numeral_numeral_real(N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N) ) ).

% numeral_less_iff
tff(fact_502_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(M)),numeral_numeral_rat(N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N) ) ).

% numeral_less_iff
tff(fact_503_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N) ) ).

% numeral_less_iff
tff(fact_504_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N) ) ).

% numeral_less_iff
tff(fact_505_vebt__succi__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xb: nat,Tb: vEBT_VEBT] : refine7594492741263601813on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_succi,Tib),Xb),aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_succi,Tb),Tib),Xb)) ).

% vebt_succi_refines
tff(fact_506_linorder__neqE__linordered__idom,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_507_linorder__neqE__linordered__idom,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_508_linorder__neqE__linordered__idom,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_509_le__num__One__iff,axiom,
    ! [Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),one)
    <=> ( Xb = one ) ) ).

% le_num_One_iff
tff(fact_510_Pair__inject,axiom,
    ! [A2: num,B2: num,A4: num,B3: num] :
      ( ( aa_num3156681643174609981um_num(product_Pair_num_num(A2),B2) = aa_num3156681643174609981um_num(product_Pair_num_num(A4),B3) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
tff(fact_511_Pair__inject,axiom,
    ! [A2: nat,B2: produc4813437837504472865T_VEBT,A4: nat,B3: produc4813437837504472865T_VEBT] :
      ( ( aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A2),B2) = aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A4),B3) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
tff(fact_512_Pair__inject,axiom,
    ! [A2: nat,B2: num,A4: nat,B3: num] :
      ( ( aa_num5026351239867140595at_num(product_Pair_nat_num(A2),B2) = aa_num5026351239867140595at_num(product_Pair_nat_num(A4),B3) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
tff(fact_513_Pair__inject,axiom,
    ! [A2: nat,B2: nat,A4: nat,B3: nat] :
      ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A4),B3) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
tff(fact_514_Pair__inject,axiom,
    ! [A2: int,B2: int,A4: int,B3: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B3) )
     => ~ ( ( A2 = A4 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
tff(fact_515_prod__cases,axiom,
    ! [P: fun_Pr6331313853612757382_num_o,P2: product_prod_num_num] :
      ( ! [A: num,B: num] : aa_Pro7685007914727650815_num_o(P,aa_num3156681643174609981um_num(product_Pair_num_num(A),B))
     => aa_Pro7685007914727650815_num_o(P,P2) ) ).

% prod_cases
tff(fact_516_prod__cases,axiom,
    ! [P: fun_Pr6320724876807202603VEBT_o,P2: produc8398139464844984134T_VEBT] :
      ( ! [A: nat,B: produc4813437837504472865T_VEBT] : aa_Pro1497393596385541012VEBT_o(P,aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A),B))
     => aa_Pro1497393596385541012VEBT_o(P,P2) ) ).

% prod_cases
tff(fact_517_prod__cases,axiom,
    ! [P: fun_Pr5555124106300582480_num_o,P2: product_prod_nat_num] :
      ( ! [A: nat,B: num] : aa_Pro6908818167415475913_num_o(P,aa_num5026351239867140595at_num(product_Pair_nat_num(A),B))
     => aa_Pro6908818167415475913_num_o(P,P2) ) ).

% prod_cases
tff(fact_518_prod__cases,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,P2: product_prod_nat_nat] :
      ( ! [A: nat,B: nat] : aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))
     => aa_Pro6150637604445964947_nat_o(P,P2) ) ).

% prod_cases
tff(fact_519_prod__cases,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,P2: product_prod_int_int] :
      ( ! [A: int,B: int] : aa_Pro4592068669816069595_int_o(P,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
     => aa_Pro4592068669816069595_int_o(P,P2) ) ).

% prod_cases
tff(fact_520_surj__pair,axiom,
    ! [P2: product_prod_num_num] :
    ? [X2: num,Y2: num] : P2 = aa_num3156681643174609981um_num(product_Pair_num_num(X2),Y2) ).

% surj_pair
tff(fact_521_surj__pair,axiom,
    ! [P2: produc8398139464844984134T_VEBT] :
    ? [X2: nat,Y2: produc4813437837504472865T_VEBT] : P2 = aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(X2),Y2) ).

% surj_pair
tff(fact_522_surj__pair,axiom,
    ! [P2: product_prod_nat_num] :
    ? [X2: nat,Y2: num] : P2 = aa_num5026351239867140595at_num(product_Pair_nat_num(X2),Y2) ).

% surj_pair
tff(fact_523_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X2: nat,Y2: nat] : P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y2) ).

% surj_pair
tff(fact_524_surj__pair,axiom,
    ! [P2: product_prod_int_int] :
    ? [X2: int,Y2: int] : P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2) ).

% surj_pair
tff(fact_525_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_num_num] :
      ~ ! [A: num,B: num] : Y != aa_num3156681643174609981um_num(product_Pair_num_num(A),B) ).

% old.prod.exhaust
tff(fact_526_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8398139464844984134T_VEBT] :
      ~ ! [A: nat,B: produc4813437837504472865T_VEBT] : Y != aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A),B) ).

% old.prod.exhaust
tff(fact_527_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_num] :
      ~ ! [A: nat,B: num] : Y != aa_num5026351239867140595at_num(product_Pair_nat_num(A),B) ).

% old.prod.exhaust
tff(fact_528_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A: nat,B: nat] : Y != aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B) ).

% old.prod.exhaust
tff(fact_529_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A: int,B: int] : Y != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) ).

% old.prod.exhaust
tff(fact_530_linorder__neqE__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_neqE_nat
tff(fact_531_infinite__descent,axiom,
    ! [P: fun_nat_o,N: nat] :
      ( ! [N3: nat] :
          ( ~ aa_nat_o(P,N3)
         => ? [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N3)
              & ~ aa_nat_o(P,M2) ) )
     => aa_nat_o(P,N) ) ).

% infinite_descent
tff(fact_532_nat__less__induct,axiom,
    ! [P: fun_nat_o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N3)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N3) )
     => aa_nat_o(P,N) ) ).

% nat_less_induct
tff(fact_533_less__irrefl__nat,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N) ).

% less_irrefl_nat
tff(fact_534_less__not__refl3,axiom,
    ! [S: nat,Tb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S),Tb)
     => ( S != Tb ) ) ).

% less_not_refl3
tff(fact_535_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
     => ( M != N ) ) ).

% less_not_refl2
tff(fact_536_less__not__refl,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N) ).

% less_not_refl
tff(fact_537_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ).

% nat_neq_iff
tff(fact_538_size__neq__size__imp__neq,axiom,
    ! [Xb: list_real,Y: list_real] :
      ( ( size_size_list_real(Xb) != size_size_list_real(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_539_size__neq__size__imp__neq,axiom,
    ! [Xb: list_o,Y: list_o] :
      ( ( size_size_list_o(Xb) != size_size_list_o(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_540_size__neq__size__imp__neq,axiom,
    ! [Xb: list_nat,Y: list_nat] :
      ( ( size_size_list_nat(Xb) != size_size_list_nat(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_541_size__neq__size__imp__neq,axiom,
    ! [Xb: list_int,Y: list_int] :
      ( ( size_size_list_int(Xb) != size_size_list_int(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_542_size__neq__size__imp__neq,axiom,
    ! [Xb: num,Y: num] :
      ( ( aa_num_nat(size_size_num,Xb) != aa_num_nat(size_size_num,Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_543_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K: nat,B2: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y2: nat] :
            ( aa_nat_o(P,Y2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),B2) )
       => ? [X2: nat] :
            ( aa_nat_o(P,X2)
            & ! [Y4: nat] :
                ( aa_nat_o(P,Y4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),X2) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_544_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ).

% nat_le_linear
tff(fact_545_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ( M = N ) ) ) ).

% le_antisym
tff(fact_546_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% eq_imp_le
tff(fact_547_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K) ) ) ).

% le_trans
tff(fact_548_le__refl,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N) ).

% le_refl
tff(fact_549_prod_Ocase__distrib,axiom,
    ! [H: fun_o_o,Fa: fun_nat_fun_nat_o,Prod: product_prod_nat_nat] :
      ( aa_o_o(H,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(Fa),Prod))
    <=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_cr(H),Fa)),Prod) ) ).

% prod.case_distrib
tff(fact_550_prod_Ocase__distrib,axiom,
    ! [H: fun_o_o,Fa: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Fa),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_cs(H),Fa)),Prod) ) ).

% prod.case_distrib
tff(fact_551_prod_Ocase__distrib,axiom,
    ! [H: fun_o_int,Fa: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_int(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Fa),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun7323606162024495668nt_int(aTP_Lamm_ct(H),Fa)),Prod) ).

% prod.case_distrib
tff(fact_552_prod_Ocase__distrib,axiom,
    ! [H: fun_int_o,Fa: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(H,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(Fa),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun82482788810740534_int_o(aTP_Lamm_cu(H),Fa)),Prod) ) ).

% prod.case_distrib
tff(fact_553_prod_Ocase__distrib,axiom,
    ! [H: fun_int_int,Fa: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int_int(H,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(Fa),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun6119123525526402106nt_int(aTP_Lamm_cv(H),Fa)),Prod) ).

% prod.case_distrib
tff(fact_554_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr4796943543331071514_nat_o,Fa: fun_na4138050610385705875at_nat,Prod: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(H,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(Fa),Prod))
    <=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_fun7288258519458182931_nat_o(aTP_Lamm_cw(H),Fa)),Prod) ) ).

% prod.case_distrib
tff(fact_555_prod_Ocase__distrib,axiom,
    ! [H: fun_o_67843798866261420at_nat,Fa: fun_nat_fun_nat_o,Prod: product_prod_nat_nat] : aa_o_P6312935399938478757at_nat(H,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(Fa),Prod)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aa_fun9180491248050787109at_nat(aTP_Lamm_cx(H),Fa)),Prod) ).

% prod.case_distrib
tff(fact_556_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr3238374608701176162_int_o,Fa: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(Fa),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_cy(H),Fa)),Prod) ) ).

% prod.case_distrib
tff(fact_557_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr982841511892225912nt_int,Fa: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(Fa),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun7461304428735181397nt_int(aTP_Lamm_cz(H),Fa)),Prod) ).

% prod.case_distrib
tff(fact_558_prod_Ocase__distrib,axiom,
    ! [H: fun_o_8988054265482332260nt_int,Fa: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_P3134438092330336605nt_int(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Fa),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_da(H),Fa)),Prod) ).

% prod.case_distrib
tff(fact_559_option_Ocase__distrib,axiom,
    ! [H: fun_num_num,F1: num,F2: fun_num_num,Option: option_num] : aa_num_num(H,aa_option_num_num(case_option_num_num(F1,F2),Option)) = aa_option_num_num(case_option_num_num(aa_num_num(H,F1),aa_fun8048995283964520608um_num(aTP_Lamm_db(H),F2)),Option) ).

% option.case_distrib
tff(fact_560_option_Ocase__distrib,axiom,
    ! [H: fun_num_int,F1: num,F2: fun_num_num,Option: option_num] : aa_num_int(H,aa_option_num_num(case_option_num_num(F1,F2),Option)) = aa_option_num_int(case_option_int_num(aa_num_int(H,F1),aa_fun3162900327506486194um_int(aTP_Lamm_dc(H),F2)),Option) ).

% option.case_distrib
tff(fact_561_option_Ocase__distrib,axiom,
    ! [H: fun_int_num,F1: int,F2: fun_num_int,Option: option_num] : aa_int_num(H,aa_option_num_int(case_option_int_num(F1,F2),Option)) = aa_option_num_num(case_option_num_num(aa_int_num(H,F1),aa_fun5215869348392759218um_num(aTP_Lamm_dd(H),F2)),Option) ).

% option.case_distrib
tff(fact_562_option_Ocase__distrib,axiom,
    ! [H: fun_int_int,F1: int,F2: fun_num_int,Option: option_num] : aa_int_int(H,aa_option_num_int(case_option_int_num(F1,F2),Option)) = aa_option_num_int(case_option_int_num(aa_int_int(H,F1),aa_fun329774391934724804um_int(aTP_Lamm_de(H),F2)),Option) ).

% option.case_distrib
tff(fact_563_option_Ocase__distrib,axiom,
    ! [H: fun_option_num_num,F1: option_num,F2: fun_num_option_num,Option: option_num] : aa_option_num_num(H,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F2),Option)) = aa_option_num_num(case_option_num_num(aa_option_num_num(H,F1),aa_fun7142274290070961008um_num(aTP_Lamm_df(H),F2)),Option) ).

% option.case_distrib
tff(fact_564_option_Ocase__distrib,axiom,
    ! [H: fun_option_num_int,F1: option_num,F2: fun_num_option_num,Option: option_num] : aa_option_num_int(H,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F2),Option)) = aa_option_num_int(case_option_int_num(aa_option_num_int(H,F1),aa_fun2256179333612926594um_int(aTP_Lamm_dg(H),F2)),Option) ).

% option.case_distrib
tff(fact_565_option_Ocase__distrib,axiom,
    ! [H: fun_num_option_num,F1: num,F2: fun_num_num,Option: option_num] : aa_num_option_num(H,aa_option_num_num(case_option_num_num(F1,F2),Option)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(H,F1),aa_fun612943910911948912on_num(aTP_Lamm_dh(H),F2)),Option) ).

% option.case_distrib
tff(fact_566_option_Ocase__distrib,axiom,
    ! [H: fun_int_option_num,F1: int,F2: fun_num_int,Option: option_num] : aa_int_option_num(H,aa_option_num_int(case_option_int_num(F1,F2),Option)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_int_option_num(H,F1),aa_fun3858197967482829954on_num(aTP_Lamm_di(H),F2)),Option) ).

% option.case_distrib
tff(fact_567_option_Ocase__distrib,axiom,
    ! [H: fun_op8989173174992673105on_num,F1: option_num,F2: fun_num_option_num,Option: option_num] : aa_opt5369159329878751634on_num(H,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F2),Option)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_opt5369159329878751634on_num(H,F1),aa_fun3636977833914307392on_num(aTP_Lamm_dj(H),F2)),Option) ).

% option.case_distrib
tff(fact_568_option_Ocase__distrib,axiom,
    ! [H: fun_o_o,F1: $o,F2: fun_Pr4796943543331071514_nat_o,Option: option4927543243414619207at_nat] :
      ( aa_o_o(H,aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((F1),F2),Option))
    <=> aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(aa_o_o(H,(F1)),aa_fun359341089595923442_nat_o(aTP_Lamm_dk(H),F2)),Option) ) ).

% option.case_distrib
tff(fact_569_le__numeral__extra_I3_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_zero_real) ).

% le_numeral_extra(3)
tff(fact_570_le__numeral__extra_I3_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_zero_rat) ).

% le_numeral_extra(3)
tff(fact_571_le__numeral__extra_I3_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),zero_zero_nat) ).

% le_numeral_extra(3)
tff(fact_572_le__numeral__extra_I3_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% le_numeral_extra(3)
tff(fact_573_less__numeral__extra_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_zero_real) ).

% less_numeral_extra(3)
tff(fact_574_less__numeral__extra_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_zero_rat) ).

% less_numeral_extra(3)
tff(fact_575_less__numeral__extra_I3_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),zero_zero_nat) ).

% less_numeral_extra(3)
tff(fact_576_less__numeral__extra_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_numeral_extra(3)
tff(fact_577_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_real != numeral_numeral_real(N) ).

% zero_neq_numeral
tff(fact_578_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_rat != numeral_numeral_rat(N) ).

% zero_neq_numeral
tff(fact_579_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,N) ).

% zero_neq_numeral
tff(fact_580_zero__neq__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_num_int(numeral_numeral_int,N) ).

% zero_neq_numeral
tff(fact_581_zero__neq__one,axiom,
    zero_zero_uint32 != one_one_uint32 ).

% zero_neq_one
tff(fact_582_zero__neq__one,axiom,
    zero_z3563351764282998399l_num1 != one_on7727431528512463931l_num1 ).

% zero_neq_one
tff(fact_583_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_584_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_585_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_586_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_587_le__numeral__extra_I4_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),one_one_real) ).

% le_numeral_extra(4)
tff(fact_588_le__numeral__extra_I4_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),one_one_rat) ).

% le_numeral_extra(4)
tff(fact_589_le__numeral__extra_I4_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),one_one_nat) ).

% le_numeral_extra(4)
tff(fact_590_le__numeral__extra_I4_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),one_one_int) ).

% le_numeral_extra(4)
tff(fact_591_less__numeral__extra_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),one_one_real) ).

% less_numeral_extra(4)
tff(fact_592_less__numeral__extra_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),one_one_rat) ).

% less_numeral_extra(4)
tff(fact_593_less__numeral__extra_I4_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),one_one_nat) ).

% less_numeral_extra(4)
tff(fact_594_less__numeral__extra_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),one_one_int) ).

% less_numeral_extra(4)
tff(fact_595_infinite__descent0,axiom,
    ! [P: fun_nat_o,N: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3)
           => ( ~ aa_nat_o(P,N3)
             => ? [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N3)
                  & ~ aa_nat_o(P,M2) ) ) )
       => aa_nat_o(P,N) ) ) ).

% infinite_descent0
tff(fact_596_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_597_less__zeroE,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),zero_zero_nat) ).

% less_zeroE
tff(fact_598_not__less0,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),zero_zero_nat) ).

% not_less0
tff(fact_599_not__gr0,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
    <=> ( N = zero_zero_nat ) ) ).

% not_gr0
tff(fact_600_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% gr0I
tff(fact_601_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),zero_zero_nat) ).

% bot_nat_0.extremum_strict
tff(fact_602_le__0__eq,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),zero_zero_nat)
    <=> ( N = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_603_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat)
     => ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_604_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat)
    <=> ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_605_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),N) ).

% less_eq_nat.simps(1)
tff(fact_606_less__mono__imp__le__mono,axiom,
    ! [Fa: fun_nat_nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(Fa,I2)),aa_nat_nat(Fa,J2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Fa,I)),aa_nat_nat(Fa,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_607_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( ( M != N )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% le_neq_implies_less
tff(fact_608_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
        | ( M = N ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% less_or_eq_imp_le
tff(fact_609_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
        | ( M = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_610_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% less_imp_le_nat
tff(fact_611_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
        & ( M != N ) ) ) ).

% nat_less_le
tff(fact_612_old_Oprod_Ocase,axiom,
    ! [Fa: fun_na4138050610385705875at_nat,X1: nat,X22: nat] : aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(Fa),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(Fa,X1),X22) ).

% old.prod.case
tff(fact_613_old_Oprod_Ocase,axiom,
    ! [Fa: fun_nat_fun_nat_o,X1: nat,X22: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(Fa),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22))
    <=> aa_nat_o(aa_nat_fun_nat_o(Fa,X1),X22) ) ).

% old.prod.case
tff(fact_614_old_Oprod_Ocase,axiom,
    ! [Fa: fun_in1696471567231710611nt_int,X1: int,X22: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Fa,X1),X22) ).

% old.prod.case
tff(fact_615_old_Oprod_Ocase,axiom,
    ! [Fa: fun_int_fun_int_o,X1: int,X22: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22))
    <=> aa_int_o(aa_int_fun_int_o(Fa,X1),X22) ) ).

% old.prod.case
tff(fact_616_old_Oprod_Ocase,axiom,
    ! [Fa: fun_int_fun_int_int,X1: int,X22: int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)) = aa_int_int(aa_int_fun_int_int(Fa,X1),X22) ).

% old.prod.case
tff(fact_617_fst__conv,axiom,
    ! [X1: num,X22: num] : aa_Pro759708478137009011um_num(product_fst_num_num,aa_num3156681643174609981um_num(product_Pair_num_num(X1),X22)) = X1 ).

% fst_conv
tff(fact_618_fst__conv,axiom,
    ! [X1: nat,X22: produc4813437837504472865T_VEBT] : aa_Pro6878622105917549268BT_nat(produc758997459209783180T_VEBT,aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(X1),X22)) = X1 ).

% fst_conv
tff(fact_619_fst__conv,axiom,
    ! [X1: nat,X22: num] : aa_Pro6261568864010571487um_nat(product_fst_nat_num,aa_num5026351239867140595at_num(product_Pair_nat_num(X1),X22)) = X1 ).

% fst_conv
tff(fact_620_fst__conv,axiom,
    ! [X1: nat,X22: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22)) = X1 ).

% fst_conv
tff(fact_621_fst__conv,axiom,
    ! [X1: int,X22: int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)) = X1 ).

% fst_conv
tff(fact_622_fst__conv,axiom,
    ! [X1: assn,X22: assn] : aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(X1),X22)) = X1 ).

% fst_conv
tff(fact_623_fst__eqD,axiom,
    ! [Xb: num,Y: num,A2: num] :
      ( ( aa_Pro759708478137009011um_num(product_fst_num_num,aa_num3156681643174609981um_num(product_Pair_num_num(Xb),Y)) = A2 )
     => ( Xb = A2 ) ) ).

% fst_eqD
tff(fact_624_fst__eqD,axiom,
    ! [Xb: nat,Y: produc4813437837504472865T_VEBT,A2: nat] :
      ( ( aa_Pro6878622105917549268BT_nat(produc758997459209783180T_VEBT,aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(Xb),Y)) = A2 )
     => ( Xb = A2 ) ) ).

% fst_eqD
tff(fact_625_fst__eqD,axiom,
    ! [Xb: nat,Y: num,A2: nat] :
      ( ( aa_Pro6261568864010571487um_nat(product_fst_nat_num,aa_num5026351239867140595at_num(product_Pair_nat_num(Xb),Y)) = A2 )
     => ( Xb = A2 ) ) ).

% fst_eqD
tff(fact_626_fst__eqD,axiom,
    ! [Xb: nat,Y: nat,A2: nat] :
      ( ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y)) = A2 )
     => ( Xb = A2 ) ) ).

% fst_eqD
tff(fact_627_fst__eqD,axiom,
    ! [Xb: int,Y: int,A2: int] :
      ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y)) = A2 )
     => ( Xb = A2 ) ) ).

% fst_eqD
tff(fact_628_fst__eqD,axiom,
    ! [Xb: assn,Y: assn,A2: assn] :
      ( ( aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(Xb),Y)) = A2 )
     => ( Xb = A2 ) ) ).

% fst_eqD
tff(fact_629_combine__options__cases,axiom,
    ! [Xb: option_nat,P: fun_op5794009454290973322_nat_o,Y: option_nat] :
      ( ( ( Xb = none_nat )
       => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) )
       => ( ! [A: nat,B: nat] :
              ( ( Xb = aa_nat_option_nat(some_nat,A) )
             => ( ( Y = aa_nat_option_nat(some_nat,B) )
               => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) ) )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_630_combine__options__cases,axiom,
    ! [Xb: option_nat,P: fun_op6120599277751319488_num_o,Y: option_num] :
      ( ( ( Xb = none_nat )
       => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) )
       => ( ! [A: nat,B: num] :
              ( ( Xb = aa_nat_option_nat(some_nat,A) )
             => ( ( Y = aa_num_option_num(some_num,B) )
               => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) ) )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_631_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op167334234089630144_nat_o,Y: option_nat] :
      ( ( ( Xb = none_num )
       => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) )
       => ( ! [A: num,B: nat] :
              ( ( Xb = aa_num_option_num(some_num,A) )
             => ( ( Y = aa_nat_option_nat(some_nat,B) )
               => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) ) )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_632_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op493924057549976310_num_o,Y: option_num] :
      ( ( ( Xb = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) )
       => ( ! [A: num,B: num] :
              ( ( Xb = aa_num_option_num(some_num,A) )
             => ( ( Y = aa_num_option_num(some_num,B) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_633_combine__options__cases,axiom,
    ! [Xb: option_nat,P: fun_op3832135881594366065_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xb = none_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) )
       => ( ! [A: nat,B: product_prod_nat_nat] :
              ( ( Xb = aa_nat_option_nat(some_nat,A) )
             => ( ( Y = some_P7363390416028606310at_nat(B) )
               => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_634_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op7326680082016885773_nat_o,Y: option_nat] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) )
       => ( ! [A: product_prod_nat_nat,B: nat] :
              ( ( Xb = some_P7363390416028606310at_nat(A) )
             => ( ( Y = aa_nat_option_nat(some_nat,B) )
               => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) ) )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_635_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op7653269905477231939_num_o,Y: option_num] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) )
       => ( ! [A: product_prod_nat_nat,B: num] :
              ( ( Xb = some_P7363390416028606310at_nat(A) )
             => ( ( Y = aa_num_option_num(some_num,B) )
               => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) ) )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_636_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op5373868528577402299_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xb = none_num )
       => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) )
       => ( ! [A: num,B: product_prod_nat_nat] :
              ( ( Xb = aa_num_option_num(some_num,A) )
             => ( ( Y = some_P7363390416028606310at_nat(B) )
               => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_637_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op1221328669798608686_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) )
       => ( ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
              ( ( Xb = some_P7363390416028606310at_nat(A) )
             => ( ( Y = some_P7363390416028606310at_nat(B) )
               => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_638_combine__options__cases,axiom,
    ! [Xb: option7339022715339332451it_nat,P: fun_op2280021900175422599_nat_o,Y: option_nat] :
      ( ( ( Xb = none_P7668321371905463026it_nat )
       => aa_option_nat_o(aa_opt9210023198028561264_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt9210023198028561264_nat_o(P,Xb),Y) )
       => ( ! [A: produc8047831477865546771it_nat,B: nat] :
              ( ( Xb = some_P468703482102919278it_nat(A) )
             => ( ( Y = aa_nat_option_nat(some_nat,B) )
               => aa_option_nat_o(aa_opt9210023198028561264_nat_o(P,Xb),Y) ) )
         => aa_option_nat_o(aa_opt9210023198028561264_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_639_split__option__all,axiom,
    ! [P: fun_op2107615217426680968_nat_o] :
      ( ! [X_12: option2621746655072343315it_nat] : aa_opt424737865643518209_nat_o(P,X_12)
    <=> ( aa_opt424737865643518209_nat_o(P,none_P1551326421579882414it_nat)
        & ! [X: produc120671012495760973it_nat] : aa_opt424737865643518209_nat_o(P,some_P2407035485129114418it_nat(X)) ) ) ).

% split_option_all
tff(fact_640_split__option__all,axiom,
    ! [P: fun_op7459662026604166478_nat_o] :
      ( ! [X_12: option7339022715339332451it_nat] : aa_opt4171472420520861111_nat_o(P,X_12)
    <=> ( aa_opt4171472420520861111_nat_o(P,none_P7668321371905463026it_nat)
        & ! [X: produc8047831477865546771it_nat] : aa_opt4171472420520861111_nat_o(P,some_P468703482102919278it_nat(X)) ) ) ).

% split_option_all
tff(fact_641_split__option__all,axiom,
    ! [P: fun_option_nat_o] :
      ( ! [X_12: option_nat] : aa_option_nat_o(P,X_12)
    <=> ( aa_option_nat_o(P,none_nat)
        & ! [X: nat] : aa_option_nat_o(P,aa_nat_option_nat(some_nat,X)) ) ) ).

% split_option_all
tff(fact_642_split__option__all,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ! [X_12: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_12)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        & ! [X: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X)) ) ) ).

% split_option_all
tff(fact_643_split__option__all,axiom,
    ! [P: fun_option_num_o] :
      ( ! [X_12: option_num] : aa_option_num_o(P,X_12)
    <=> ( aa_option_num_o(P,none_num)
        & ! [X: num] : aa_option_num_o(P,aa_num_option_num(some_num,X)) ) ) ).

% split_option_all
tff(fact_644_split__option__ex,axiom,
    ! [P: fun_op2107615217426680968_nat_o] :
      ( ? [X_12: option2621746655072343315it_nat] : aa_opt424737865643518209_nat_o(P,X_12)
    <=> ( aa_opt424737865643518209_nat_o(P,none_P1551326421579882414it_nat)
        | ? [X: produc120671012495760973it_nat] : aa_opt424737865643518209_nat_o(P,some_P2407035485129114418it_nat(X)) ) ) ).

% split_option_ex
tff(fact_645_split__option__ex,axiom,
    ! [P: fun_op7459662026604166478_nat_o] :
      ( ? [X_12: option7339022715339332451it_nat] : aa_opt4171472420520861111_nat_o(P,X_12)
    <=> ( aa_opt4171472420520861111_nat_o(P,none_P7668321371905463026it_nat)
        | ? [X: produc8047831477865546771it_nat] : aa_opt4171472420520861111_nat_o(P,some_P468703482102919278it_nat(X)) ) ) ).

% split_option_ex
tff(fact_646_split__option__ex,axiom,
    ! [P: fun_option_nat_o] :
      ( ? [X_12: option_nat] : aa_option_nat_o(P,X_12)
    <=> ( aa_option_nat_o(P,none_nat)
        | ? [X: nat] : aa_option_nat_o(P,aa_nat_option_nat(some_nat,X)) ) ) ).

% split_option_ex
tff(fact_647_split__option__ex,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ? [X_12: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_12)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        | ? [X: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X)) ) ) ).

% split_option_ex
tff(fact_648_split__option__ex,axiom,
    ! [P: fun_option_num_o] :
      ( ? [X_12: option_num] : aa_option_num_o(P,X_12)
    <=> ( aa_option_num_o(P,none_num)
        | ? [X: num] : aa_option_num_o(P,aa_num_option_num(some_num,X)) ) ) ).

% split_option_ex
tff(fact_649_option_Oexhaust,axiom,
    ! [Y: option2621746655072343315it_nat] :
      ( ( Y != none_P1551326421579882414it_nat )
     => ~ ! [X23: produc120671012495760973it_nat] : Y != some_P2407035485129114418it_nat(X23) ) ).

% option.exhaust
tff(fact_650_option_Oexhaust,axiom,
    ! [Y: option7339022715339332451it_nat] :
      ( ( Y != none_P7668321371905463026it_nat )
     => ~ ! [X23: produc8047831477865546771it_nat] : Y != some_P468703482102919278it_nat(X23) ) ).

% option.exhaust
tff(fact_651_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] : Y != aa_nat_option_nat(some_nat,X23) ) ).

% option.exhaust
tff(fact_652_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X23) ) ).

% option.exhaust
tff(fact_653_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] : Y != aa_num_option_num(some_num,X23) ) ).

% option.exhaust
tff(fact_654_option_OdiscI,axiom,
    ! [Option: option2621746655072343315it_nat,X22: produc120671012495760973it_nat] :
      ( ( Option = some_P2407035485129114418it_nat(X22) )
     => ( Option != none_P1551326421579882414it_nat ) ) ).

% option.discI
tff(fact_655_option_OdiscI,axiom,
    ! [Option: option7339022715339332451it_nat,X22: produc8047831477865546771it_nat] :
      ( ( Option = some_P468703482102919278it_nat(X22) )
     => ( Option != none_P7668321371905463026it_nat ) ) ).

% option.discI
tff(fact_656_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option = aa_nat_option_nat(some_nat,X22) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_657_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X22) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_658_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option = aa_num_option_num(some_num,X22) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_659_option_Odistinct_I1_J,axiom,
    ! [X22: produc120671012495760973it_nat] : none_P1551326421579882414it_nat != some_P2407035485129114418it_nat(X22) ).

% option.distinct(1)
tff(fact_660_option_Odistinct_I1_J,axiom,
    ! [X22: produc8047831477865546771it_nat] : none_P7668321371905463026it_nat != some_P468703482102919278it_nat(X22) ).

% option.distinct(1)
tff(fact_661_option_Odistinct_I1_J,axiom,
    ! [X22: nat] : none_nat != aa_nat_option_nat(some_nat,X22) ).

% option.distinct(1)
tff(fact_662_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X22) ).

% option.distinct(1)
tff(fact_663_option_Odistinct_I1_J,axiom,
    ! [X22: num] : none_num != aa_num_option_num(some_num,X22) ).

% option.distinct(1)
tff(fact_664_snd__conv,axiom,
    ! [X1: num,X22: num] : aa_Pro759708478137009011um_num(product_snd_num_num,aa_num3156681643174609981um_num(product_Pair_num_num(X1),X22)) = X22 ).

% snd_conv
tff(fact_665_snd__conv,axiom,
    ! [X1: nat,X22: produc4813437837504472865T_VEBT] : aa_Pro3664079603483438877T_VEBT(produc2084898568784432842T_VEBT,aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(X1),X22)) = X22 ).

% snd_conv
tff(fact_666_snd__conv,axiom,
    ! [X1: nat,X22: num] : aa_Pro2818900650166350121um_num(product_snd_nat_num,aa_num5026351239867140595at_num(product_Pair_nat_num(X1),X22)) = X22 ).

% snd_conv
tff(fact_667_snd__conv,axiom,
    ! [X1: nat,X22: nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X1),X22)) = X22 ).

% snd_conv
tff(fact_668_snd__conv,axiom,
    ! [X1: int,X22: int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X22)) = X22 ).

% snd_conv
tff(fact_669_snd__conv,axiom,
    ! [X1: assn,X22: assn] : aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(X1),X22)) = X22 ).

% snd_conv
tff(fact_670_snd__eqD,axiom,
    ! [Xb: num,Y: num,A2: num] :
      ( ( aa_Pro759708478137009011um_num(product_snd_num_num,aa_num3156681643174609981um_num(product_Pair_num_num(Xb),Y)) = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
tff(fact_671_snd__eqD,axiom,
    ! [Xb: nat,Y: produc4813437837504472865T_VEBT,A2: produc4813437837504472865T_VEBT] :
      ( ( aa_Pro3664079603483438877T_VEBT(produc2084898568784432842T_VEBT,aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(Xb),Y)) = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
tff(fact_672_snd__eqD,axiom,
    ! [Xb: nat,Y: num,A2: num] :
      ( ( aa_Pro2818900650166350121um_num(product_snd_nat_num,aa_num5026351239867140595at_num(product_Pair_nat_num(Xb),Y)) = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
tff(fact_673_snd__eqD,axiom,
    ! [Xb: nat,Y: nat,A2: nat] :
      ( ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y)) = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
tff(fact_674_snd__eqD,axiom,
    ! [Xb: int,Y: int,A2: int] :
      ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y)) = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
tff(fact_675_snd__eqD,axiom,
    ! [Xb: assn,Y: assn,A2: assn] :
      ( ( aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(Xb),Y)) = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
tff(fact_676_prod__induct3,axiom,
    ! [P: fun_Pr6320724876807202603VEBT_o,Xb: produc8398139464844984134T_VEBT] :
      ( ! [A: nat,B: list_VEBT_VEBT,C2: vEBT_VEBT] : aa_Pro1497393596385541012VEBT_o(P,aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A),produc6691630295680060561T_VEBT(B,C2)))
     => aa_Pro1497393596385541012VEBT_o(P,Xb) ) ).

% prod_induct3
tff(fact_677_prod__cases3,axiom,
    ! [Y: produc8398139464844984134T_VEBT] :
      ~ ! [A: nat,B: list_VEBT_VEBT,C2: vEBT_VEBT] : Y != aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A),produc6691630295680060561T_VEBT(B,C2)) ).

% prod_cases3
tff(fact_678_option_Osel,axiom,
    ! [X22: nat] : the_nat(aa_nat_option_nat(some_nat,X22)) = X22 ).

% option.sel
tff(fact_679_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X22)) = X22 ).

% option.sel
tff(fact_680_option_Osel,axiom,
    ! [X22: num] : aa_option_num_num(the_num,aa_num_option_num(some_num,X22)) = X22 ).

% option.sel
tff(fact_681_prod__eq__iff,axiom,
    ! [S: product_prod_nat_nat,Tb: product_prod_nat_nat] :
      ( ( S = Tb )
    <=> ( ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,S) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Tb) )
        & ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,S) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Tb) ) ) ) ).

% prod_eq_iff
tff(fact_682_prod__eq__iff,axiom,
    ! [S: product_prod_int_int,Tb: product_prod_int_int] :
      ( ( S = Tb )
    <=> ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,S) = aa_Pro6799258434465214121nt_int(product_fst_int_int,Tb) )
        & ( aa_Pro6799258434465214121nt_int(product_snd_int_int,S) = aa_Pro6799258434465214121nt_int(product_snd_int_int,Tb) ) ) ) ).

% prod_eq_iff
tff(fact_683_prod__eq__iff,axiom,
    ! [S: produc6575502325842934193n_assn,Tb: produc6575502325842934193n_assn] :
      ( ( S = Tb )
    <=> ( ( aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,S) = aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Tb) )
        & ( aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,S) = aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Tb) ) ) ) ).

% prod_eq_iff
tff(fact_684_prod__eqI,axiom,
    ! [P2: product_prod_nat_nat,Q2: product_prod_nat_nat] :
      ( ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Q2) )
     => ( ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Q2) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
tff(fact_685_prod__eqI,axiom,
    ! [P2: product_prod_int_int,Q2: product_prod_int_int] :
      ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,P2) = aa_Pro6799258434465214121nt_int(product_fst_int_int,Q2) )
     => ( ( aa_Pro6799258434465214121nt_int(product_snd_int_int,P2) = aa_Pro6799258434465214121nt_int(product_snd_int_int,Q2) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
tff(fact_686_prod__eqI,axiom,
    ! [P2: produc6575502325842934193n_assn,Q2: produc6575502325842934193n_assn] :
      ( ( aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,P2) = aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Q2) )
     => ( ( aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,P2) = aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Q2) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
tff(fact_687_prod_Oexpand,axiom,
    ! [Prod: product_prod_nat_nat,Prod2: product_prod_nat_nat] :
      ( ( ( aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod2) )
        & ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_688_prod_Oexpand,axiom,
    ! [Prod: product_prod_int_int,Prod2: product_prod_int_int] :
      ( ( ( aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod) = aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod2) )
        & ( aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod) = aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_689_prod_Oexpand,axiom,
    ! [Prod: produc6575502325842934193n_assn,Prod2: produc6575502325842934193n_assn] :
      ( ( ( aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Prod) = aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Prod2) )
        & ( aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Prod) = aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_690_option_Oexpand,axiom,
    ! [Option: option2621746655072343315it_nat,Option2: option2621746655072343315it_nat] :
      ( ( ( Option = none_P1551326421579882414it_nat )
      <=> ( Option2 = none_P1551326421579882414it_nat ) )
     => ( ( ( Option != none_P1551326421579882414it_nat )
         => ( ( Option2 != none_P1551326421579882414it_nat )
           => ( the_Pr3501439614016493281it_nat(Option) = the_Pr3501439614016493281it_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_691_option_Oexpand,axiom,
    ! [Option: option7339022715339332451it_nat,Option2: option7339022715339332451it_nat] :
      ( ( ( Option = none_P7668321371905463026it_nat )
      <=> ( Option2 = none_P7668321371905463026it_nat ) )
     => ( ( ( Option != none_P7668321371905463026it_nat )
         => ( ( Option2 != none_P7668321371905463026it_nat )
           => ( the_Pr5838048819577852031it_nat(Option) = the_Pr5838048819577852031it_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_692_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
tff(fact_693_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
      <=> ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( aa_option_num_num(the_num,Option) = aa_option_num_num(the_num,Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_694_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
tff(fact_695_option_Osimps_I5_J,axiom,
    ! [F1: heap_Time_Heap_o,F2: fun_Pr1329037631615965750Heap_o,X22: product_prod_nat_nat] : case_o1442776274061689234at_nat(F1,F2,some_P7363390416028606310at_nat(X22)) = aa_Pro691548381575413351Heap_o(F2,X22) ).

% option.simps(5)
tff(fact_696_option_Osimps_I5_J,axiom,
    ! [F1: option_num,F2: fun_num_option_num,X22: num] : aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F2),aa_num_option_num(some_num,X22)) = aa_num_option_num(F2,X22) ).

% option.simps(5)
tff(fact_697_option_Osimps_I5_J,axiom,
    ! [F1: $o,F2: fun_Pr4796943543331071514_nat_o,X22: product_prod_nat_nat] :
      ( aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((F1),F2),some_P7363390416028606310at_nat(X22))
    <=> aa_Pro6150637604445964947_nat_o(F2,X22) ) ).

% option.simps(5)
tff(fact_698_option_Osimps_I5_J,axiom,
    ! [F1: num,F2: fun_num_num,X22: num] : aa_option_num_num(case_option_num_num(F1,F2),aa_num_option_num(some_num,X22)) = aa_num_num(F2,X22) ).

% option.simps(5)
tff(fact_699_option_Osimps_I5_J,axiom,
    ! [F1: int,F2: fun_num_int,X22: num] : aa_option_num_int(case_option_int_num(F1,F2),aa_num_option_num(some_num,X22)) = aa_num_int(F2,X22) ).

% option.simps(5)
tff(fact_700_option_Osimps_I4_J,axiom,
    ! [F1: heap_Time_Heap_o,F2: fun_Pr1329037631615965750Heap_o] : case_o1442776274061689234at_nat(F1,F2,none_P5556105721700978146at_nat) = F1 ).

% option.simps(4)
tff(fact_701_option_Osimps_I4_J,axiom,
    ! [F1: option_num,F2: fun_num_option_num] : aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F2),none_num) = F1 ).

% option.simps(4)
tff(fact_702_option_Osimps_I4_J,axiom,
    ! [F1: $o,F2: fun_Pr4796943543331071514_nat_o] :
      ( aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((F1),F2),none_P5556105721700978146at_nat)
    <=> (F1) ) ).

% option.simps(4)
tff(fact_703_option_Osimps_I4_J,axiom,
    ! [F1: num,F2: fun_num_num] : aa_option_num_num(case_option_num_num(F1,F2),none_num) = F1 ).

% option.simps(4)
tff(fact_704_option_Osimps_I4_J,axiom,
    ! [F1: int,F2: fun_num_int] : aa_option_num_int(case_option_int_num(F1,F2),none_num) = F1 ).

% option.simps(4)
tff(fact_705_cond__case__prod__eta,axiom,
    ! [Fa: fun_na4138050610385705875at_nat,G: fun_Pr2118491707810114563at_nat] :
      ( ! [X2: nat,Y2: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(Fa,X2),Y2) = aa_Pro262675715343573244at_nat(G,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y2))
     => ( produc2626176000494625587at_nat(Fa) = G ) ) ).

% cond_case_prod_eta
tff(fact_706_cond__case__prod__eta,axiom,
    ! [Fa: fun_nat_fun_nat_o,G: fun_Pr4796943543331071514_nat_o] :
      ( ! [X2: nat,Y2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(Fa,X2),Y2)
        <=> aa_Pro6150637604445964947_nat_o(G,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y2)) )
     => ( produc6081775807080527818_nat_o(Fa) = G ) ) ).

% cond_case_prod_eta
tff(fact_707_cond__case__prod__eta,axiom,
    ! [Fa: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
      ( ! [X2: int,Y2: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Fa,X2),Y2) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2))
     => ( produc4245557441103728435nt_int(Fa) = G ) ) ).

% cond_case_prod_eta
tff(fact_708_cond__case__prod__eta,axiom,
    ! [Fa: fun_int_fun_int_o,G: fun_Pr3238374608701176162_int_o] :
      ( ! [X2: int,Y2: int] :
          ( aa_int_o(aa_int_fun_int_o(Fa,X2),Y2)
        <=> aa_Pro4592068669816069595_int_o(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2)) )
     => ( produc4947309494688390418_int_o(Fa) = G ) ) ).

% cond_case_prod_eta
tff(fact_709_cond__case__prod__eta,axiom,
    ! [Fa: fun_int_fun_int_int,G: fun_Pr982841511892225912nt_int] :
      ( ! [X2: int,Y2: int] : aa_int_int(aa_int_fun_int_int(Fa,X2),Y2) = aa_Pro6799258434465214121nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2))
     => ( produc8211389475949308722nt_int(Fa) = G ) ) ).

% cond_case_prod_eta
tff(fact_710_case__prod__eta,axiom,
    ! [Fa: fun_Pr2118491707810114563at_nat] : produc2626176000494625587at_nat(aTP_Lamm_dl(Fa)) = Fa ).

% case_prod_eta
tff(fact_711_case__prod__eta,axiom,
    ! [Fa: fun_Pr4796943543331071514_nat_o] : produc6081775807080527818_nat_o(aTP_Lamm_dm(Fa)) = Fa ).

% case_prod_eta
tff(fact_712_case__prod__eta,axiom,
    ! [Fa: fun_Pr2478283820680522243nt_int] : produc4245557441103728435nt_int(aTP_Lamm_dn(Fa)) = Fa ).

% case_prod_eta
tff(fact_713_case__prod__eta,axiom,
    ! [Fa: fun_Pr3238374608701176162_int_o] : produc4947309494688390418_int_o(aTP_Lamm_do(Fa)) = Fa ).

% case_prod_eta
tff(fact_714_case__prod__eta,axiom,
    ! [Fa: fun_Pr982841511892225912nt_int] : produc8211389475949308722nt_int(aTP_Lamm_dp(Fa)) = Fa ).

% case_prod_eta
tff(fact_715_case__prodE2,axiom,
    ! [Q: fun_Pr4796943543331071514_nat_o,P: fun_na4138050610385705875at_nat,Z: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(Q,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(P),Z))
     => ~ ! [X2: nat,Y2: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y2) )
           => ~ aa_Pro6150637604445964947_nat_o(Q,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(P,X2),Y2)) ) ) ).

% case_prodE2
tff(fact_716_case__prodE2,axiom,
    ! [Q: fun_o_o,P: fun_nat_fun_nat_o,Z: product_prod_nat_nat] :
      ( aa_o_o(Q,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(P),Z))
     => ~ ! [X2: nat,Y2: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y2) )
           => ~ aa_o_o(Q,aa_nat_o(aa_nat_fun_nat_o(P,X2),Y2)) ) ) ).

% case_prodE2
tff(fact_717_case__prodE2,axiom,
    ! [Q: fun_Pr3238374608701176162_int_o,P: fun_in1696471567231710611nt_int,Z: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Q,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(P),Z))
     => ~ ! [X2: int,Y2: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2) )
           => ~ aa_Pro4592068669816069595_int_o(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(P,X2),Y2)) ) ) ).

% case_prodE2
tff(fact_718_case__prodE2,axiom,
    ! [Q: fun_o_o,P: fun_int_fun_int_o,Z: product_prod_int_int] :
      ( aa_o_o(Q,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(P),Z))
     => ~ ! [X2: int,Y2: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2) )
           => ~ aa_o_o(Q,aa_int_o(aa_int_fun_int_o(P,X2),Y2)) ) ) ).

% case_prodE2
tff(fact_719_case__prodE2,axiom,
    ! [Q: fun_int_o,P: fun_int_fun_int_int,Z: product_prod_int_int] :
      ( aa_int_o(Q,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(P),Z))
     => ~ ! [X2: int,Y2: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2) )
           => ~ aa_int_o(Q,aa_int_int(aa_int_fun_int_int(P,X2),Y2)) ) ) ).

% case_prodE2
tff(fact_720_fst__def,axiom,
    ! [Prod: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_dq),Prod) ).

% fst_def
tff(fact_721_fst__def,axiom,
    ! [Prod: produc6575502325842934193n_assn] : aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Prod) = aa_Pro2979818503705714077n_assn(produc2152611005075324454n_assn(aTP_Lamm_dr),Prod) ).

% fst_def
tff(fact_722_fst__def,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aTP_Lamm_ds),Prod) ).

% fst_def
tff(fact_723_snd__def,axiom,
    ! [Prod: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_dt),Prod) ).

% snd_def
tff(fact_724_snd__def,axiom,
    ! [Prod: produc6575502325842934193n_assn] : aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Prod) = aa_Pro2979818503705714077n_assn(produc2152611005075324454n_assn(aTP_Lamm_du),Prod) ).

% snd_def
tff(fact_725_snd__def,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aTP_Lamm_dv),Prod) ).

% snd_def
tff(fact_726_not__numeral__le__zero,axiom,
    ! [N: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(N)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_727_not__numeral__le__zero,axiom,
    ! [N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(N)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_728_not__numeral__le__zero,axiom,
    ! [N: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_729_not__numeral__le__zero,axiom,
    ! [N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_730_zero__le__numeral,axiom,
    ! [N: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),numeral_numeral_real(N)) ).

% zero_le_numeral
tff(fact_731_zero__le__numeral,axiom,
    ! [N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(N)) ).

% zero_le_numeral
tff(fact_732_zero__le__numeral,axiom,
    ! [N: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N)) ).

% zero_le_numeral
tff(fact_733_zero__le__numeral,axiom,
    ! [N: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,N)) ).

% zero_le_numeral
tff(fact_734_not__numeral__less__zero,axiom,
    ! [N: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(N)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_735_not__numeral__less__zero,axiom,
    ! [N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(N)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_736_not__numeral__less__zero,axiom,
    ! [N: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_737_not__numeral__less__zero,axiom,
    ! [N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_738_zero__less__numeral,axiom,
    ! [N: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),numeral_numeral_real(N)) ).

% zero_less_numeral
tff(fact_739_zero__less__numeral,axiom,
    ! [N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),numeral_numeral_rat(N)) ).

% zero_less_numeral
tff(fact_740_zero__less__numeral,axiom,
    ! [N: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,N)) ).

% zero_less_numeral
tff(fact_741_zero__less__numeral,axiom,
    ! [N: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,N)) ).

% zero_less_numeral
tff(fact_742_not__one__le__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),zero_zero_real) ).

% not_one_le_zero
tff(fact_743_not__one__le__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),zero_zero_rat) ).

% not_one_le_zero
tff(fact_744_not__one__le__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),zero_zero_nat) ).

% not_one_le_zero
tff(fact_745_not__one__le__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),zero_zero_int) ).

% not_one_le_zero
tff(fact_746_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_747_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_748_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_749_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_750_zero__less__one__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% zero_less_one_class.zero_le_one
tff(fact_751_zero__less__one__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one_class.zero_le_one
tff(fact_752_zero__less__one__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one_class.zero_le_one
tff(fact_753_zero__less__one__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% zero_less_one_class.zero_le_one
tff(fact_754_not__one__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),zero_zero_real) ).

% not_one_less_zero
tff(fact_755_not__one__less__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),zero_zero_rat) ).

% not_one_less_zero
tff(fact_756_not__one__less__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),zero_zero_nat) ).

% not_one_less_zero
tff(fact_757_not__one__less__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),zero_zero_int) ).

% not_one_less_zero
tff(fact_758_zero__less__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% zero_less_one
tff(fact_759_zero__less__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one
tff(fact_760_zero__less__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one
tff(fact_761_zero__less__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% zero_less_one
tff(fact_762_less__numeral__extra_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% less_numeral_extra(1)
tff(fact_763_less__numeral__extra_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% less_numeral_extra(1)
tff(fact_764_less__numeral__extra_I1_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% less_numeral_extra(1)
tff(fact_765_less__numeral__extra_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% less_numeral_extra(1)
tff(fact_766_one__le__numeral,axiom,
    ! [N: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),numeral_numeral_real(N)) ).

% one_le_numeral
tff(fact_767_one__le__numeral,axiom,
    ! [N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),numeral_numeral_rat(N)) ).

% one_le_numeral
tff(fact_768_one__le__numeral,axiom,
    ! [N: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) ).

% one_le_numeral
tff(fact_769_one__le__numeral,axiom,
    ! [N: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,N)) ).

% one_le_numeral
tff(fact_770_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(N)),one_one_real) ).

% not_numeral_less_one
tff(fact_771_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(N)),one_one_rat) ).

% not_numeral_less_one
tff(fact_772_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat) ).

% not_numeral_less_one
tff(fact_773_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N)),one_one_int) ).

% not_numeral_less_one
tff(fact_774_numeral__One,axiom,
    numera9087168376688890119uint32(one) = one_one_uint32 ).

% numeral_One
tff(fact_775_numeral__One,axiom,
    numera7442385471795722001l_num1(one) = one_on7727431528512463931l_num1 ).

% numeral_One
tff(fact_776_numeral__One,axiom,
    numeral_numeral_real(one) = one_one_real ).

% numeral_One
tff(fact_777_numeral__One,axiom,
    numeral_numeral_rat(one) = one_one_rat ).

% numeral_One
tff(fact_778_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_779_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_780_divide__numeral__1,axiom,
    ! [A2: real] : aa_real_real(divide_divide_real(A2),numeral_numeral_real(one)) = A2 ).

% divide_numeral_1
tff(fact_781_divide__numeral__1,axiom,
    ! [A2: rat] : divide_divide_rat(A2,numeral_numeral_rat(one)) = A2 ).

% divide_numeral_1
tff(fact_782_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_783_ex__least__nat__le,axiom,
    ! [P: fun_nat_o,N: nat] :
      ( aa_nat_o(P,N)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N)
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
               => ~ aa_nat_o(P,I3) )
            & aa_nat_o(P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_784_surjective__pairing,axiom,
    ! [Tb: product_prod_num_num] : Tb = aa_num3156681643174609981um_num(product_Pair_num_num(aa_Pro759708478137009011um_num(product_fst_num_num,Tb)),aa_Pro759708478137009011um_num(product_snd_num_num,Tb)) ).

% surjective_pairing
tff(fact_785_surjective__pairing,axiom,
    ! [Tb: produc8398139464844984134T_VEBT] : Tb = aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(aa_Pro6878622105917549268BT_nat(produc758997459209783180T_VEBT,Tb)),aa_Pro3664079603483438877T_VEBT(produc2084898568784432842T_VEBT,Tb)) ).

% surjective_pairing
tff(fact_786_surjective__pairing,axiom,
    ! [Tb: product_prod_nat_num] : Tb = aa_num5026351239867140595at_num(product_Pair_nat_num(aa_Pro6261568864010571487um_nat(product_fst_nat_num,Tb)),aa_Pro2818900650166350121um_num(product_snd_nat_num,Tb)) ).

% surjective_pairing
tff(fact_787_surjective__pairing,axiom,
    ! [Tb: product_prod_nat_nat] : Tb = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Tb)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Tb)) ).

% surjective_pairing
tff(fact_788_surjective__pairing,axiom,
    ! [Tb: product_prod_int_int] : Tb = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Tb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Tb)) ).

% surjective_pairing
tff(fact_789_surjective__pairing,axiom,
    ! [Tb: produc6575502325842934193n_assn] : Tb = aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Tb)),aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Tb)) ).

% surjective_pairing
tff(fact_790_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_num_num] : Prod = aa_num3156681643174609981um_num(product_Pair_num_num(aa_Pro759708478137009011um_num(product_fst_num_num,Prod)),aa_Pro759708478137009011um_num(product_snd_num_num,Prod)) ).

% prod.exhaust_sel
tff(fact_791_prod_Oexhaust__sel,axiom,
    ! [Prod: produc8398139464844984134T_VEBT] : Prod = aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(aa_Pro6878622105917549268BT_nat(produc758997459209783180T_VEBT,Prod)),aa_Pro3664079603483438877T_VEBT(produc2084898568784432842T_VEBT,Prod)) ).

% prod.exhaust_sel
tff(fact_792_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_num] : Prod = aa_num5026351239867140595at_num(product_Pair_nat_num(aa_Pro6261568864010571487um_nat(product_fst_nat_num,Prod)),aa_Pro2818900650166350121um_num(product_snd_nat_num,Prod)) ).

% prod.exhaust_sel
tff(fact_793_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_nat] : Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) ).

% prod.exhaust_sel
tff(fact_794_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_int_int] : Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ).

% prod.exhaust_sel
tff(fact_795_prod_Oexhaust__sel,axiom,
    ! [Prod: produc6575502325842934193n_assn] : Prod = aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Prod)),aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Prod)) ).

% prod.exhaust_sel
tff(fact_796_option_Oexhaust__sel,axiom,
    ! [Option: option2621746655072343315it_nat] :
      ( ( Option != none_P1551326421579882414it_nat )
     => ( Option = some_P2407035485129114418it_nat(the_Pr3501439614016493281it_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_797_option_Oexhaust__sel,axiom,
    ! [Option: option7339022715339332451it_nat] :
      ( ( Option != none_P7668321371905463026it_nat )
     => ( Option = some_P468703482102919278it_nat(the_Pr5838048819577852031it_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_798_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = aa_nat_option_nat(some_nat,the_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_799_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_800_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) ) ) ).

% option.exhaust_sel
tff(fact_801_case__prod__beta,axiom,
    ! [Fa: fun_na4138050610385705875at_nat,P2: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(Fa),P2) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(Fa,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2)) ).

% case_prod_beta
tff(fact_802_case__prod__beta,axiom,
    ! [Fa: fun_nat_fun_nat_o,P2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(Fa),P2)
    <=> aa_nat_o(aa_nat_fun_nat_o(Fa,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2)) ) ).

% case_prod_beta
tff(fact_803_case__prod__beta,axiom,
    ! [Fa: fun_in1696471567231710611nt_int,P2: product_prod_int_int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(Fa),P2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Fa,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)) ).

% case_prod_beta
tff(fact_804_case__prod__beta,axiom,
    ! [Fa: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Fa),P2)
    <=> aa_int_o(aa_int_fun_int_o(Fa,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)) ) ).

% case_prod_beta
tff(fact_805_case__prod__beta,axiom,
    ! [Fa: fun_int_fun_int_int,P2: product_prod_int_int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(Fa),P2) = aa_int_int(aa_int_fun_int_int(Fa,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)) ).

% case_prod_beta
tff(fact_806_split__beta,axiom,
    ! [Fa: fun_na4138050610385705875at_nat,Prod: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(Fa),Prod) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(Fa,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) ).

% split_beta
tff(fact_807_split__beta,axiom,
    ! [Fa: fun_nat_fun_nat_o,Prod: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(Fa),Prod)
    <=> aa_nat_o(aa_nat_fun_nat_o(Fa,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) ) ).

% split_beta
tff(fact_808_split__beta,axiom,
    ! [Fa: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(Fa),Prod) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Fa,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ).

% split_beta
tff(fact_809_split__beta,axiom,
    ! [Fa: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Fa),Prod)
    <=> aa_int_o(aa_int_fun_int_o(Fa,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ) ).

% split_beta
tff(fact_810_split__beta,axiom,
    ! [Fa: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(Fa),Prod) = aa_int_int(aa_int_fun_int_int(Fa,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) ).

% split_beta
tff(fact_811_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_812_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_813_int__div__less__self,axiom,
    ! [Xb: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),K)),Xb) ) ) ).

% int_div_less_self
tff(fact_814_div__neg__pos__less0,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),zero_zero_int) ) ) ).

% div_neg_pos_less0
tff(fact_815_option_Ocase__eq__if,axiom,
    ! [F1: heap_Time_Heap_o,F2: fun_Pr1329037631615965750Heap_o,Option: option4927543243414619207at_nat] :
      case_o1442776274061689234at_nat(F1,F2,Option) = $ite(Option = none_P5556105721700978146at_nat,F1,aa_Pro691548381575413351Heap_o(F2,the_Pr8591224930841456533at_nat(Option))) ).

% option.case_eq_if
tff(fact_816_option_Ocase__eq__if,axiom,
    ! [F1: option_num,F2: fun_num_option_num,Option: option_num] :
      aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F2),Option) = $ite(Option = none_num,F1,aa_num_option_num(F2,aa_option_num_num(the_num,Option))) ).

% option.case_eq_if
tff(fact_817_option_Ocase__eq__if,axiom,
    ! [F1: $o,F2: fun_Pr4796943543331071514_nat_o,Option: option4927543243414619207at_nat] :
      ( aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((F1),F2),Option)
    <=> $ite(Option = none_P5556105721700978146at_nat,(F1),aa_Pro6150637604445964947_nat_o(F2,the_Pr8591224930841456533at_nat(Option))) ) ).

% option.case_eq_if
tff(fact_818_option_Ocase__eq__if,axiom,
    ! [F1: num,F2: fun_num_num,Option: option_num] :
      aa_option_num_num(case_option_num_num(F1,F2),Option) = $ite(Option = none_num,F1,aa_num_num(F2,aa_option_num_num(the_num,Option))) ).

% option.case_eq_if
tff(fact_819_option_Ocase__eq__if,axiom,
    ! [F1: int,F2: fun_num_int,Option: option_num] :
      aa_option_num_int(case_option_int_num(F1,F2),Option) = $ite(Option = none_num,F1,aa_num_int(F2,aa_option_num_num(the_num,Option))) ).

% option.case_eq_if
tff(fact_820_case__prod__unfold,axiom,
    ! [X3: fun_na4138050610385705875at_nat,Xa: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(X3),Xa) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(X3,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa)) ).

% case_prod_unfold
tff(fact_821_case__prod__unfold,axiom,
    ! [X3: fun_nat_fun_nat_o,Xa: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(X3),Xa)
    <=> aa_nat_o(aa_nat_fun_nat_o(X3,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xa)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xa)) ) ).

% case_prod_unfold
tff(fact_822_case__prod__unfold,axiom,
    ! [X3: fun_in1696471567231710611nt_int,Xa: product_prod_int_int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(X3),Xa) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(X3,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)) ).

% case_prod_unfold
tff(fact_823_case__prod__unfold,axiom,
    ! [X3: fun_int_fun_int_o,Xa: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(X3),Xa)
    <=> aa_int_o(aa_int_fun_int_o(X3,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)) ) ).

% case_prod_unfold
tff(fact_824_case__prod__unfold,axiom,
    ! [X3: fun_int_fun_int_int,Xa: product_prod_int_int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(X3),Xa) = aa_int_int(aa_int_fun_int_int(X3,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xa)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xa)) ).

% case_prod_unfold
tff(fact_825_case__prod__beta_H,axiom,
    ! [Fa: fun_na4138050610385705875at_nat,X3: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(Fa),X3) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(Fa,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,X3)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,X3)) ).

% case_prod_beta'
tff(fact_826_case__prod__beta_H,axiom,
    ! [Fa: fun_nat_fun_nat_o,X3: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(Fa),X3)
    <=> aa_nat_o(aa_nat_fun_nat_o(Fa,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,X3)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,X3)) ) ).

% case_prod_beta'
tff(fact_827_case__prod__beta_H,axiom,
    ! [Fa: fun_in1696471567231710611nt_int,X3: product_prod_int_int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(Fa),X3) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Fa,aa_Pro6799258434465214121nt_int(product_fst_int_int,X3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3)) ).

% case_prod_beta'
tff(fact_828_case__prod__beta_H,axiom,
    ! [Fa: fun_int_fun_int_o,X3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Fa),X3)
    <=> aa_int_o(aa_int_fun_int_o(Fa,aa_Pro6799258434465214121nt_int(product_fst_int_int,X3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3)) ) ).

% case_prod_beta'
tff(fact_829_case__prod__beta_H,axiom,
    ! [Fa: fun_int_fun_int_int,X3: product_prod_int_int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(Fa),X3) = aa_int_int(aa_int_fun_int_int(Fa,aa_Pro6799258434465214121nt_int(product_fst_int_int,X3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,X3)) ).

% case_prod_beta'
tff(fact_830_option_Othe__def,axiom,
    ! [Option: option6094739215472523721uint32] : the_fu3257833805496833637uint32(Option) = case_o1863524157202330495uint32(undefi5883245521155754897uint32,aTP_Lamm_dw,Option) ).

% option.the_def
tff(fact_831_option_Othe__def,axiom,
    ! [Option: option3423447615326530758eger_o] : the_fu7763965076112332130eger_o(Option) = case_o2694340534094624761eger_o(undefi9030187962712549518eger_o,aTP_Lamm_dx,Option) ).

% option.the_def
tff(fact_832_option_Othe__def,axiom,
    ! [Option: option5345528073251348614uint32] : the_fu4420311046302572578uint32(Option) = case_o61842737470731385uint32(undefi8935238026407452494uint32,aTP_Lamm_dy,Option) ).

% option.the_def
tff(fact_833_option_Othe__def,axiom,
    ! [Option: option3426696969397718942uint32] : the_fu1630297056608640954uint32(Option) = case_o5757947405571428393uint32(undefi424500106705823974uint32,aTP_Lamm_dz,Option) ).

% option.the_def
tff(fact_834_option_Othe__def,axiom,
    ! [Option: option4768052937140784030uint32] : the_fu5356795541574441914uint32(Option) = case_o3262472310887385129uint32(undefi4150998591671624934uint32,aTP_Lamm_ea,Option) ).

% option.the_def
tff(fact_835_option_Othe__def,axiom,
    ! [Option: option_nat] : the_nat(Option) = case_option_nat_nat(undefined_nat,aTP_Lamm_eb,Option) ).

% option.the_def
tff(fact_836_option_Othe__def,axiom,
    ! [Option: option4927543243414619207at_nat] : the_Pr8591224930841456533at_nat(Option) = case_o7430979018509204427at_nat(undefi3946296454836805481at_nat,aTP_Lamm_ec,Option) ).

% option.the_def
tff(fact_837_option_Othe__def,axiom,
    ! [Option: option_num] : aa_option_num_num(the_num,Option) = aa_option_num_num(case_option_num_num(undefined_num,aTP_Lamm_ed),Option) ).

% option.the_def
tff(fact_838_div__positive,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ) ) ).

% div_positive
tff(fact_839_div__positive,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% div_positive
tff(fact_840_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_841_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_842_prod_Osplit__sel__asm,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Fa: fun_na4138050610385705875at_nat,Prod: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(P,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(Fa),Prod))
    <=> ~ ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
          & ~ aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(Fa,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_843_prod_Osplit__sel__asm,axiom,
    ! [P: fun_o_o,Fa: fun_nat_fun_nat_o,Prod: product_prod_nat_nat] :
      ( aa_o_o(P,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(Fa),Prod))
    <=> ~ ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
          & ~ aa_o_o(P,aa_nat_o(aa_nat_fun_nat_o(Fa,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_844_prod_Osplit__sel__asm,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Fa: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(P,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(Fa),Prod))
    <=> ~ ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
          & ~ aa_Pro4592068669816069595_int_o(P,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Fa,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_845_prod_Osplit__sel__asm,axiom,
    ! [P: fun_o_o,Fa: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(P,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Fa),Prod))
    <=> ~ ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
          & ~ aa_o_o(P,aa_int_o(aa_int_fun_int_o(Fa,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_846_prod_Osplit__sel__asm,axiom,
    ! [P: fun_int_o,Fa: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(P,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(Fa),Prod))
    <=> ~ ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
          & ~ aa_int_o(P,aa_int_int(aa_int_fun_int_int(Fa,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_847_prod_Osplit__sel,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Fa: fun_na4138050610385705875at_nat,Prod: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(P,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(Fa),Prod))
    <=> ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
       => aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(Fa,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel
tff(fact_848_prod_Osplit__sel,axiom,
    ! [P: fun_o_o,Fa: fun_nat_fun_nat_o,Prod: product_prod_nat_nat] :
      ( aa_o_o(P,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(Fa),Prod))
    <=> ( ( Prod = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod)) )
       => aa_o_o(P,aa_nat_o(aa_nat_fun_nat_o(Fa,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Prod)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Prod))) ) ) ).

% prod.split_sel
tff(fact_849_prod_Osplit__sel,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Fa: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(P,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(Fa),Prod))
    <=> ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
       => aa_Pro4592068669816069595_int_o(P,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Fa,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel
tff(fact_850_prod_Osplit__sel,axiom,
    ! [P: fun_o_o,Fa: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(P,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Fa),Prod))
    <=> ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
       => aa_o_o(P,aa_int_o(aa_int_fun_int_o(Fa,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel
tff(fact_851_prod_Osplit__sel,axiom,
    ! [P: fun_int_o,Fa: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(P,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(Fa),Prod))
    <=> ( ( Prod = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod)) )
       => aa_int_o(P,aa_int_int(aa_int_fun_int_int(Fa,aa_Pro6799258434465214121nt_int(product_fst_int_int,Prod)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Prod))) ) ) ).

% prod.split_sel
tff(fact_852_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_853_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_854_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_855_zdiv__le__dividend,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),A2) ) ) ).

% zdiv_le_dividend
tff(fact_856_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_857_div__nonpos__pos__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_858_div__nonneg__neg__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_859_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)) ) ) ).

% div_positive_int
tff(fact_860_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ) ).

% div_int_pos_iff
tff(fact_861_zdiv__mono2__neg,axiom,
    ! [A2: int,B3: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_862_zdiv__mono1__neg,axiom,
    ! [A2: int,A4: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),A4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% zdiv_mono1_neg
tff(fact_863_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_864_zdiv__mono2,axiom,
    ! [A2: int,B3: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B3)) ) ) ) ).

% zdiv_mono2
tff(fact_865_zdiv__mono1,axiom,
    ! [A2: int,A4: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),A4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),B2)) ) ) ).

% zdiv_mono1
tff(fact_866_option_Osplit__sel__asm,axiom,
    ! [P: fun_He8247548780604731983ap_o_o,F1: heap_Time_Heap_o,F2: fun_Pr1329037631615965750Heap_o,Option: option4927543243414619207at_nat] :
      ( aa_Hea4522166450600650424ap_o_o(P,case_o1442776274061689234at_nat(F1,F2,Option))
    <=> ~ ( ( ( Option = none_P5556105721700978146at_nat )
            & ~ aa_Hea4522166450600650424ap_o_o(P,F1) )
          | ( ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) )
            & ~ aa_Hea4522166450600650424ap_o_o(P,aa_Pro691548381575413351Heap_o(F2,the_Pr8591224930841456533at_nat(Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_867_option_Osplit__sel__asm,axiom,
    ! [P: fun_option_num_o,F1: option_num,F2: fun_num_option_num,Option: option_num] :
      ( aa_option_num_o(P,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F2),Option))
    <=> ~ ( ( ( Option = none_num )
            & ~ aa_option_num_o(P,F1) )
          | ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
            & ~ aa_option_num_o(P,aa_num_option_num(F2,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_868_option_Osplit__sel__asm,axiom,
    ! [P: fun_o_o,F1: $o,F2: fun_Pr4796943543331071514_nat_o,Option: option4927543243414619207at_nat] :
      ( aa_o_o(P,aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((F1),F2),Option))
    <=> ~ ( ( ( Option = none_P5556105721700978146at_nat )
            & ~ aa_o_o(P,(F1)) )
          | ( ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) )
            & ~ aa_o_o(P,aa_Pro6150637604445964947_nat_o(F2,the_Pr8591224930841456533at_nat(Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_869_option_Osplit__sel__asm,axiom,
    ! [P: fun_num_o,F1: num,F2: fun_num_num,Option: option_num] :
      ( aa_num_o(P,aa_option_num_num(case_option_num_num(F1,F2),Option))
    <=> ~ ( ( ( Option = none_num )
            & ~ aa_num_o(P,F1) )
          | ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
            & ~ aa_num_o(P,aa_num_num(F2,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_870_option_Osplit__sel__asm,axiom,
    ! [P: fun_int_o,F1: int,F2: fun_num_int,Option: option_num] :
      ( aa_int_o(P,aa_option_num_int(case_option_int_num(F1,F2),Option))
    <=> ~ ( ( ( Option = none_num )
            & ~ aa_int_o(P,F1) )
          | ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
            & ~ aa_int_o(P,aa_num_int(F2,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_871_option_Osplit__sel,axiom,
    ! [P: fun_He8247548780604731983ap_o_o,F1: heap_Time_Heap_o,F2: fun_Pr1329037631615965750Heap_o,Option: option4927543243414619207at_nat] :
      ( aa_Hea4522166450600650424ap_o_o(P,case_o1442776274061689234at_nat(F1,F2,Option))
    <=> ( ( ( Option = none_P5556105721700978146at_nat )
         => aa_Hea4522166450600650424ap_o_o(P,F1) )
        & ( ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) )
         => aa_Hea4522166450600650424ap_o_o(P,aa_Pro691548381575413351Heap_o(F2,the_Pr8591224930841456533at_nat(Option))) ) ) ) ).

% option.split_sel
tff(fact_872_option_Osplit__sel,axiom,
    ! [P: fun_option_num_o,F1: option_num,F2: fun_num_option_num,Option: option_num] :
      ( aa_option_num_o(P,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(F1,F2),Option))
    <=> ( ( ( Option = none_num )
         => aa_option_num_o(P,F1) )
        & ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
         => aa_option_num_o(P,aa_num_option_num(F2,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel
tff(fact_873_option_Osplit__sel,axiom,
    ! [P: fun_o_o,F1: $o,F2: fun_Pr4796943543331071514_nat_o,Option: option4927543243414619207at_nat] :
      ( aa_o_o(P,aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((F1),F2),Option))
    <=> ( ( ( Option = none_P5556105721700978146at_nat )
         => aa_o_o(P,(F1)) )
        & ( ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) )
         => aa_o_o(P,aa_Pro6150637604445964947_nat_o(F2,the_Pr8591224930841456533at_nat(Option))) ) ) ) ).

% option.split_sel
tff(fact_874_option_Osplit__sel,axiom,
    ! [P: fun_num_o,F1: num,F2: fun_num_num,Option: option_num] :
      ( aa_num_o(P,aa_option_num_num(case_option_num_num(F1,F2),Option))
    <=> ( ( ( Option = none_num )
         => aa_num_o(P,F1) )
        & ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
         => aa_num_o(P,aa_num_num(F2,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel
tff(fact_875_option_Osplit__sel,axiom,
    ! [P: fun_int_o,F1: int,F2: fun_num_int,Option: option_num] :
      ( aa_int_o(P,aa_option_num_int(case_option_int_num(F1,F2),Option))
    <=> ( ( ( Option = none_num )
         => aa_int_o(P,F1) )
        & ( ( Option = aa_num_option_num(some_num,aa_option_num_num(the_num,Option)) )
         => aa_int_o(P,aa_num_int(F2,aa_option_num_num(the_num,Option))) ) ) ) ).

% option.split_sel
tff(fact_876_half__gt__zero__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A2),numeral_numeral_real(aa_num_num(bit0,one))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2) ) ).

% half_gt_zero_iff
tff(fact_877_half__gt__zero__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(A2,numeral_numeral_rat(aa_num_num(bit0,one))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ).

% half_gt_zero_iff
tff(fact_878_half__gt__zero,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A2),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_879_half__gt__zero,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),divide_divide_rat(A2,numeral_numeral_rat(aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_880_semiring__norm_I76_J,axiom,
    ! [N: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,N)) ).

% semiring_norm(76)
tff(fact_881_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),one) ).

% semiring_norm(69)
tff(fact_882_pred__list__to__short,axiom,
    ! [Deg2: nat,Xb: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Ma)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),Xb) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_883_VEBT__internal_Ovebt__succi_H_Osimps,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_succi,Tb),Tib),Xb) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_eu(Tb),Xb),aTP_Lamm_cq(Xb)),Tib) ).

% VEBT_internal.vebt_succi'.simps
tff(fact_884_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,M),one) ).

% semiring_norm(75)
tff(fact_885_semiring__norm_I68_J,axiom,
    ! [N: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),N) ).

% semiring_norm(68)
tff(fact_886_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit0,N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N) ) ).

% semiring_norm(78)
tff(fact_887_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit0,N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N) ) ).

% semiring_norm(71)
tff(fact_888_pred__max,axiom,
    ! [Deg2: nat,Ma: nat,Xb: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xb)
       => ( aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),Xb) = aa_nat_option_nat(some_nat,Ma) ) ) ) ).

% pred_max
tff(fact_889_less__option__None__Some__code,axiom,
    ! [Xb: nat] : aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,none_nat),aa_nat_option_nat(some_nat,Xb)) ).

% less_option_None_Some_code
tff(fact_890_less__option__None__Some__code,axiom,
    ! [Xb: num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,none_num),aa_num_option_num(some_num,Xb)) ).

% less_option_None_Some_code
tff(fact_891_less__eq__option__Some__None,axiom,
    ! [Xb: nat] : ~ aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,aa_nat_option_nat(some_nat,Xb)),none_nat) ).

% less_eq_option_Some_None
tff(fact_892_less__eq__option__Some__None,axiom,
    ! [Xb: num] : ~ aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,aa_num_option_num(some_num,Xb)),none_num) ).

% less_eq_option_Some_None
tff(fact_893_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) ) ).

% enat_ord_number(1)
tff(fact_894_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_num(bit0,M) = aa_num_num(bit0,N) )
    <=> ( M = N ) ) ).

% semiring_norm(87)
tff(fact_895_case__prodI,axiom,
    ! [Fa: fun_num_fun_num_o,A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(Fa,A2),B2)
     => aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(Fa),aa_num3156681643174609981um_num(product_Pair_num_num(A2),B2)) ) ).

% case_prodI
tff(fact_896_case__prodI,axiom,
    ! [Fa: fun_na5540905182366533063VEBT_o,A2: nat,B2: produc4813437837504472865T_VEBT] :
      ( aa_Pro3461009536670225267VEBT_o(aa_nat6863640065291604488VEBT_o(Fa,A2),B2)
     => aa_Pro1497393596385541012VEBT_o(produc2834603712688810931VEBT_o(Fa),aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A2),B2)) ) ).

% case_prodI
tff(fact_897_case__prodI,axiom,
    ! [Fa: fun_nat_fun_num_o,A2: nat,B2: num] :
      ( aa_num_o(aa_nat_fun_num_o(Fa,A2),B2)
     => aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(Fa),aa_num5026351239867140595at_num(product_Pair_nat_num(A2),B2)) ) ).

% case_prodI
tff(fact_898_case__prodI,axiom,
    ! [Fa: fun_nat_fun_nat_o,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(Fa,A2),B2)
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(Fa),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2)) ) ).

% case_prodI
tff(fact_899_case__prodI,axiom,
    ! [Fa: fun_int_fun_int_o,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(Fa,A2),B2)
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2)) ) ).

% case_prodI
tff(fact_900_case__prodI2,axiom,
    ! [P2: product_prod_num_num,C: fun_num_fun_num_o] :
      ( ! [A: num,B: num] :
          ( ( P2 = aa_num3156681643174609981um_num(product_Pair_num_num(A),B) )
         => aa_num_o(aa_num_fun_num_o(C,A),B) )
     => aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(C),P2) ) ).

% case_prodI2
tff(fact_901_case__prodI2,axiom,
    ! [P2: produc8398139464844984134T_VEBT,C: fun_na5540905182366533063VEBT_o] :
      ( ! [A: nat,B: produc4813437837504472865T_VEBT] :
          ( ( P2 = aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A),B) )
         => aa_Pro3461009536670225267VEBT_o(aa_nat6863640065291604488VEBT_o(C,A),B) )
     => aa_Pro1497393596385541012VEBT_o(produc2834603712688810931VEBT_o(C),P2) ) ).

% case_prodI2
tff(fact_902_case__prodI2,axiom,
    ! [P2: product_prod_nat_num,C: fun_nat_fun_num_o] :
      ( ! [A: nat,B: num] :
          ( ( P2 = aa_num5026351239867140595at_num(product_Pair_nat_num(A),B) )
         => aa_num_o(aa_nat_fun_num_o(C,A),B) )
     => aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(C),P2) ) ).

% case_prodI2
tff(fact_903_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: fun_nat_fun_nat_o] :
      ( ! [A: nat,B: nat] :
          ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B) )
         => aa_nat_o(aa_nat_fun_nat_o(C,A),B) )
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2) ) ).

% case_prodI2
tff(fact_904_case__prodI2,axiom,
    ! [P2: product_prod_int_int,C: fun_int_fun_int_o] :
      ( ! [A: int,B: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) )
         => aa_int_o(aa_int_fun_int_o(C,A),B) )
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2) ) ).

% case_prodI2
tff(fact_905_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_nu530747996811530292et_nat,A2: num,B2: num] :
      ( aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A2),B2))
     => aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,aa_num3156681643174609981um_num(product_Pair_num_num(A2),B2))) ) ).

% mem_case_prodI
tff(fact_906_mem__case__prodI,axiom,
    ! [Z: vEBT_VEBT,C: fun_nu1671606405228480914T_VEBT,A2: num,B2: num] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_num_set_VEBT_VEBT(aa_num1587092468593290619T_VEBT(C,A2),B2))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc1023323404773863986T_VEBT(C,aa_num3156681643174609981um_num(product_Pair_num_num(A2),B2))) ) ).

% mem_case_prodI
tff(fact_907_mem__case__prodI,axiom,
    ! [Z: int,C: fun_nu4940571614878387344et_int,A2: num,B2: num] :
      ( aa_set_int_o(member_int(Z),aa_num_set_int(aa_num3127423267984116305et_int(C,A2),B2))
     => aa_set_int_o(member_int(Z),produc6406642877701697732et_int(C,aa_num3156681643174609981um_num(product_Pair_num_num(A2),B2))) ) ).

% mem_case_prodI
tff(fact_908_mem__case__prodI,axiom,
    ! [Z: real,C: fun_nu4698968662094489488t_real,A2: num,B2: num] :
      ( aa_set_real_o(member_real(Z),aa_num_set_real(aa_num2469771427700027217t_real(C,A2),B2))
     => aa_set_real_o(member_real(Z),produc8296048397933160132t_real(C,aa_num3156681643174609981um_num(product_Pair_num_num(A2),B2))) ) ).

% mem_case_prodI
tff(fact_909_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_na3896347130304379370et_nat,A2: nat,B2: num] :
      ( aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_nat8983105546408692139et_nat(C,A2),B2))
     => aa_set_nat_o(member_nat(Z),produc4130284055270567454et_nat(C,aa_num5026351239867140595at_num(product_Pair_nat_num(A2),B2))) ) ).

% mem_case_prodI
tff(fact_910_mem__case__prodI,axiom,
    ! [Z: vEBT_VEBT,C: fun_na5890587069057688924T_VEBT,A2: nat,B2: num] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_num_set_VEBT_VEBT(aa_nat5806073132422498629T_VEBT(C,A2),B2))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc7751347746043978236T_VEBT(C,aa_num5026351239867140595at_num(product_Pair_nat_num(A2),B2))) ) ).

% mem_case_prodI
tff(fact_911_mem__case__prodI,axiom,
    ! [Z: int,C: fun_na8306170748371236422et_int,A2: nat,B2: num] :
      ( aa_set_int_o(member_int(Z),aa_num_set_int(aa_nat6493022401476965383et_int(C,A2),B2))
     => aa_set_int_o(member_int(Z),produc9175805072616146554et_int(C,aa_num5026351239867140595at_num(product_Pair_nat_num(A2),B2))) ) ).

% mem_case_prodI
tff(fact_912_mem__case__prodI,axiom,
    ! [Z: real,C: fun_na3666575854997362246t_real,A2: nat,B2: num] :
      ( aa_set_real_o(member_real(Z),aa_num_set_real(aa_nat1437378620602899975t_real(C,A2),B2))
     => aa_set_real_o(member_real(Z),produc1435849484188172666t_real(C,aa_num5026351239867140595at_num(product_Pair_nat_num(A2),B2))) ) ).

% mem_case_prodI
tff(fact_913_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_na6128946301132549280et_nat,A2: nat,B2: nat] :
      ( aa_set_nat_o(member_nat(Z),aa_nat_set_nat(aa_nat1818925681583257441et_nat(C,A2),B2))
     => aa_set_nat_o(member_nat(Z),produc6189476227299908564et_nat(C,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2))) ) ).

% mem_case_prodI
tff(fact_914_mem__case__prodI,axiom,
    ! [Z: vEBT_VEBT,C: fun_na1247094492991354918T_VEBT,A2: nat,B2: nat] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_nat_set_VEBT_VEBT(aa_nat5764631470541234447T_VEBT(C,A2),B2))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc7709906084162714054T_VEBT(C,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2))) ) ).

% mem_case_prodI
tff(fact_915_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: nat,C: fun_nu530747996811530292et_nat] :
      ( ! [A: num,B: num] :
          ( ( P2 = aa_num3156681643174609981um_num(product_Pair_num_num(A),B) )
         => aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A),B)) )
     => aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_916_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: vEBT_VEBT,C: fun_nu1671606405228480914T_VEBT] :
      ( ! [A: num,B: num] :
          ( ( P2 = aa_num3156681643174609981um_num(product_Pair_num_num(A),B) )
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_num_set_VEBT_VEBT(aa_num1587092468593290619T_VEBT(C,A),B)) )
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc1023323404773863986T_VEBT(C,P2)) ) ).

% mem_case_prodI2
tff(fact_917_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: int,C: fun_nu4940571614878387344et_int] :
      ( ! [A: num,B: num] :
          ( ( P2 = aa_num3156681643174609981um_num(product_Pair_num_num(A),B) )
         => aa_set_int_o(member_int(Z),aa_num_set_int(aa_num3127423267984116305et_int(C,A),B)) )
     => aa_set_int_o(member_int(Z),produc6406642877701697732et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_918_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: real,C: fun_nu4698968662094489488t_real] :
      ( ! [A: num,B: num] :
          ( ( P2 = aa_num3156681643174609981um_num(product_Pair_num_num(A),B) )
         => aa_set_real_o(member_real(Z),aa_num_set_real(aa_num2469771427700027217t_real(C,A),B)) )
     => aa_set_real_o(member_real(Z),produc8296048397933160132t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_919_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: nat,C: fun_na3896347130304379370et_nat] :
      ( ! [A: nat,B: num] :
          ( ( P2 = aa_num5026351239867140595at_num(product_Pair_nat_num(A),B) )
         => aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_nat8983105546408692139et_nat(C,A),B)) )
     => aa_set_nat_o(member_nat(Z),produc4130284055270567454et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_920_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: vEBT_VEBT,C: fun_na5890587069057688924T_VEBT] :
      ( ! [A: nat,B: num] :
          ( ( P2 = aa_num5026351239867140595at_num(product_Pair_nat_num(A),B) )
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_num_set_VEBT_VEBT(aa_nat5806073132422498629T_VEBT(C,A),B)) )
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc7751347746043978236T_VEBT(C,P2)) ) ).

% mem_case_prodI2
tff(fact_921_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: int,C: fun_na8306170748371236422et_int] :
      ( ! [A: nat,B: num] :
          ( ( P2 = aa_num5026351239867140595at_num(product_Pair_nat_num(A),B) )
         => aa_set_int_o(member_int(Z),aa_num_set_int(aa_nat6493022401476965383et_int(C,A),B)) )
     => aa_set_int_o(member_int(Z),produc9175805072616146554et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_922_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: real,C: fun_na3666575854997362246t_real] :
      ( ! [A: nat,B: num] :
          ( ( P2 = aa_num5026351239867140595at_num(product_Pair_nat_num(A),B) )
         => aa_set_real_o(member_real(Z),aa_num_set_real(aa_nat1437378620602899975t_real(C,A),B)) )
     => aa_set_real_o(member_real(Z),produc1435849484188172666t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_923_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: nat,C: fun_na6128946301132549280et_nat] :
      ( ! [A: nat,B: nat] :
          ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B) )
         => aa_set_nat_o(member_nat(Z),aa_nat_set_nat(aa_nat1818925681583257441et_nat(C,A),B)) )
     => aa_set_nat_o(member_nat(Z),produc6189476227299908564et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_924_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: vEBT_VEBT,C: fun_na1247094492991354918T_VEBT] :
      ( ! [A: nat,B: nat] :
          ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B) )
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_nat_set_VEBT_VEBT(aa_nat5764631470541234447T_VEBT(C,A),B)) )
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc7709906084162714054T_VEBT(C,P2)) ) ).

% mem_case_prodI2
tff(fact_925_semiring__norm_I85_J,axiom,
    ! [M: num] : aa_num_num(bit0,M) != one ).

% semiring_norm(85)
tff(fact_926_semiring__norm_I83_J,axiom,
    ! [N: num] : one != aa_num_num(bit0,N) ).

% semiring_norm(83)
tff(fact_927_less__eq__option__None__code,axiom,
    ! [Xb: option_num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,none_num),Xb) ).

% less_eq_option_None_code
tff(fact_928_less__eq__option__None__code,axiom,
    ! [Xb: option_nat] : aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,none_nat),Xb) ).

% less_eq_option_None_code
tff(fact_929_succ__min,axiom,
    ! [Deg2: nat,Xb: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi)
       => ( aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),Xb) = aa_nat_option_nat(some_nat,Mi) ) ) ) ).

% succ_min
tff(fact_930_less__option__None,axiom,
    ! [Xb: option_num] : ~ aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,Xb),none_num) ).

% less_option_None
tff(fact_931_less__option__None,axiom,
    ! [Xb: option_nat] : ~ aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,Xb),none_nat) ).

% less_option_None
tff(fact_932_succ__list__to__short,axiom,
    ! [Deg2: nat,Mi: nat,Xb: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),Xb) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_933_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) ) ).

% enat_ord_number(2)
tff(fact_934_less__eq__option__Some,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ord_le2843612097646854710et_nat(some_set_nat(Xb),some_set_nat(Y))
    <=> aa_set_nat_o(ord_less_eq_set_nat(Xb),Y) ) ).

% less_eq_option_Some
tff(fact_935_less__eq__option__Some,axiom,
    ! [Xb: rat,Y: rat] :
      ( ord_le2406147912482264968on_rat(some_rat(Xb),some_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% less_eq_option_Some
tff(fact_936_less__eq__option__Some,axiom,
    ! [Xb: num,Y: num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,aa_num_option_num(some_num,Xb)),aa_num_option_num(some_num,Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ).

% less_eq_option_Some
tff(fact_937_less__eq__option__Some,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,aa_nat_option_nat(some_nat,Xb)),aa_nat_option_nat(some_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% less_eq_option_Some
tff(fact_938_less__eq__option__Some,axiom,
    ! [Xb: int,Y: int] :
      ( ord_le1736525451366464988on_int(aa_int_option_int(some_int,Xb),aa_int_option_int(some_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% less_eq_option_Some
tff(fact_939_less__option__Some,axiom,
    ! [Xb: real,Y: real] :
      ( ord_less_option_real(some_real(Xb),some_real(Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% less_option_Some
tff(fact_940_less__option__Some,axiom,
    ! [Xb: rat,Y: rat] :
      ( ord_less_option_rat(some_rat(Xb),some_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).

% less_option_Some
tff(fact_941_less__option__Some,axiom,
    ! [Xb: num,Y: num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,aa_num_option_num(some_num,Xb)),aa_num_option_num(some_num,Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y) ) ).

% less_option_Some
tff(fact_942_less__option__Some,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,aa_nat_option_nat(some_nat,Xb)),aa_nat_option_nat(some_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ).

% less_option_Some
tff(fact_943_less__option__Some,axiom,
    ! [Xb: int,Y: int] :
      ( ord_less_option_int(aa_int_option_int(some_int,Xb),aa_int_option_int(some_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ).

% less_option_Some
tff(fact_944_succ__greatereq__min,axiom,
    ! [Deg2: nat,Mi: nat,Xb: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Xb)
       => ( aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),Xb) = $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h)),l)),
                      $let(
                        sc: option_nat,
                        sc:= aa_nat_option_nat(vEBT_vebt_succ(Summary2),h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_945_succ__less__length__list,axiom,
    ! [Deg2: nat,Mi: nat,Xb: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
         => ( aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),Xb) = $let(
                l: nat,
                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h)),l)),
                      $let(
                        sc: option_nat,
                        sc:= aa_nat_option_nat(vEBT_vebt_succ(Summary2),h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_946_pred__less__length__list,axiom,
    ! [Deg2: nat,Xb: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Ma)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
         => ( aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),Xb) = $let(
                l: nat,
                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h)),l)),
                      $let(
                        pr: option_nat,
                        pr:= aa_nat_option_nat(vEBT_vebt_pred(Summary2),h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xb),aa_nat_option_nat(some_nat,Mi),none_nat),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_947_pred__lesseq__max,axiom,
    ! [Deg2: nat,Xb: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Ma)
       => ( aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),Xb) = $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h)),l)),
                      $let(
                        pr: option_nat,
                        pr:= aa_nat_option_nat(vEBT_vebt_pred(Summary2),h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xb),aa_nat_option_nat(some_nat,Mi),none_nat),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_948_i0__lb,axiom,
    ! [N: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),N) ).

% i0_lb
tff(fact_949_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(N),zero_z5237406670263579293d_enat)
    <=> ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_950_less__eq__option__def,axiom,
    ! [Xb: option_set_nat,Y: option_set_nat] :
      ( ord_le2843612097646854710et_nat(Xb,Y)
    <=> case_o4401850862724306899et_nat($true,aTP_Lamm_ev(Y),Xb) ) ).

% less_eq_option_def
tff(fact_951_less__eq__option__def,axiom,
    ! [Xb: option_rat,Y: option_rat] :
      ( ord_le2406147912482264968on_rat(Xb,Y)
    <=> case_option_o_rat($true,aTP_Lamm_ew(Y),Xb) ) ).

% less_eq_option_def
tff(fact_952_less__eq__option__def,axiom,
    ! [Xb: option_num,Y: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,Xb),Y)
    <=> aa_option_num_o(case_option_o_num($true,aTP_Lamm_ex(Y)),Xb) ) ).

% less_eq_option_def
tff(fact_953_less__eq__option__def,axiom,
    ! [Xb: option_nat,Y: option_nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,Xb),Y)
    <=> aa_option_nat_o(case_option_o_nat($true,aTP_Lamm_ey(Y)),Xb) ) ).

% less_eq_option_def
tff(fact_954_less__eq__option__def,axiom,
    ! [Xb: option_int,Y: option_int] :
      ( ord_le1736525451366464988on_int(Xb,Y)
    <=> case_option_o_int($true,aTP_Lamm_ez(Y),Xb) ) ).

% less_eq_option_def
tff(fact_955_less__option__def,axiom,
    ! [Xb: option_real,Y: option_real] :
      ( ord_less_option_real(Xb,Y)
    <=> case_option_o_real($false,aTP_Lamm_fb(Xb),Y) ) ).

% less_option_def
tff(fact_956_less__option__def,axiom,
    ! [Xb: option_rat,Y: option_rat] :
      ( ord_less_option_rat(Xb,Y)
    <=> case_option_o_rat($false,aTP_Lamm_fd(Xb),Y) ) ).

% less_option_def
tff(fact_957_less__option__def,axiom,
    ! [Xb: option_num,Y: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,Xb),Y)
    <=> aa_option_num_o(case_option_o_num($false,aTP_Lamm_ff(Xb)),Y) ) ).

% less_option_def
tff(fact_958_less__option__def,axiom,
    ! [Xb: option_nat,Y: option_nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,Xb),Y)
    <=> aa_option_nat_o(case_option_o_nat($false,aTP_Lamm_fh(Xb)),Y) ) ).

% less_option_def
tff(fact_959_less__option__def,axiom,
    ! [Xb: option_int,Y: option_int] :
      ( ord_less_option_int(Xb,Y)
    <=> case_option_o_int($false,aTP_Lamm_fj(Xb),Y) ) ).

% less_option_def
tff(fact_960_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_nu530747996811530292et_nat,P2: product_prod_num_num] :
      ( aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,P2))
     => ~ ! [X2: num,Y2: num] :
            ( ( P2 = aa_num3156681643174609981um_num(product_Pair_num_num(X2),Y2) )
           => ~ aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_961_mem__case__prodE,axiom,
    ! [Z: vEBT_VEBT,C: fun_nu1671606405228480914T_VEBT,P2: product_prod_num_num] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc1023323404773863986T_VEBT(C,P2))
     => ~ ! [X2: num,Y2: num] :
            ( ( P2 = aa_num3156681643174609981um_num(product_Pair_num_num(X2),Y2) )
           => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_num_set_VEBT_VEBT(aa_num1587092468593290619T_VEBT(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_962_mem__case__prodE,axiom,
    ! [Z: int,C: fun_nu4940571614878387344et_int,P2: product_prod_num_num] :
      ( aa_set_int_o(member_int(Z),produc6406642877701697732et_int(C,P2))
     => ~ ! [X2: num,Y2: num] :
            ( ( P2 = aa_num3156681643174609981um_num(product_Pair_num_num(X2),Y2) )
           => ~ aa_set_int_o(member_int(Z),aa_num_set_int(aa_num3127423267984116305et_int(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_963_mem__case__prodE,axiom,
    ! [Z: real,C: fun_nu4698968662094489488t_real,P2: product_prod_num_num] :
      ( aa_set_real_o(member_real(Z),produc8296048397933160132t_real(C,P2))
     => ~ ! [X2: num,Y2: num] :
            ( ( P2 = aa_num3156681643174609981um_num(product_Pair_num_num(X2),Y2) )
           => ~ aa_set_real_o(member_real(Z),aa_num_set_real(aa_num2469771427700027217t_real(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_964_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_na3896347130304379370et_nat,P2: product_prod_nat_num] :
      ( aa_set_nat_o(member_nat(Z),produc4130284055270567454et_nat(C,P2))
     => ~ ! [X2: nat,Y2: num] :
            ( ( P2 = aa_num5026351239867140595at_num(product_Pair_nat_num(X2),Y2) )
           => ~ aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_nat8983105546408692139et_nat(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_965_mem__case__prodE,axiom,
    ! [Z: vEBT_VEBT,C: fun_na5890587069057688924T_VEBT,P2: product_prod_nat_num] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc7751347746043978236T_VEBT(C,P2))
     => ~ ! [X2: nat,Y2: num] :
            ( ( P2 = aa_num5026351239867140595at_num(product_Pair_nat_num(X2),Y2) )
           => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_num_set_VEBT_VEBT(aa_nat5806073132422498629T_VEBT(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_966_mem__case__prodE,axiom,
    ! [Z: int,C: fun_na8306170748371236422et_int,P2: product_prod_nat_num] :
      ( aa_set_int_o(member_int(Z),produc9175805072616146554et_int(C,P2))
     => ~ ! [X2: nat,Y2: num] :
            ( ( P2 = aa_num5026351239867140595at_num(product_Pair_nat_num(X2),Y2) )
           => ~ aa_set_int_o(member_int(Z),aa_num_set_int(aa_nat6493022401476965383et_int(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_967_mem__case__prodE,axiom,
    ! [Z: real,C: fun_na3666575854997362246t_real,P2: product_prod_nat_num] :
      ( aa_set_real_o(member_real(Z),produc1435849484188172666t_real(C,P2))
     => ~ ! [X2: nat,Y2: num] :
            ( ( P2 = aa_num5026351239867140595at_num(product_Pair_nat_num(X2),Y2) )
           => ~ aa_set_real_o(member_real(Z),aa_num_set_real(aa_nat1437378620602899975t_real(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_968_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_na6128946301132549280et_nat,P2: product_prod_nat_nat] :
      ( aa_set_nat_o(member_nat(Z),produc6189476227299908564et_nat(C,P2))
     => ~ ! [X2: nat,Y2: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y2) )
           => ~ aa_set_nat_o(member_nat(Z),aa_nat_set_nat(aa_nat1818925681583257441et_nat(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_969_mem__case__prodE,axiom,
    ! [Z: vEBT_VEBT,C: fun_na1247094492991354918T_VEBT,P2: product_prod_nat_nat] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),produc7709906084162714054T_VEBT(C,P2))
     => ~ ! [X2: nat,Y2: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y2) )
           => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z),aa_nat_set_VEBT_VEBT(aa_nat5764631470541234447T_VEBT(C,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_970_case__prodD,axiom,
    ! [Fa: fun_num_fun_num_o,A2: num,B2: num] :
      ( aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(Fa),aa_num3156681643174609981um_num(product_Pair_num_num(A2),B2))
     => aa_num_o(aa_num_fun_num_o(Fa,A2),B2) ) ).

% case_prodD
tff(fact_971_case__prodD,axiom,
    ! [Fa: fun_na5540905182366533063VEBT_o,A2: nat,B2: produc4813437837504472865T_VEBT] :
      ( aa_Pro1497393596385541012VEBT_o(produc2834603712688810931VEBT_o(Fa),aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A2),B2))
     => aa_Pro3461009536670225267VEBT_o(aa_nat6863640065291604488VEBT_o(Fa,A2),B2) ) ).

% case_prodD
tff(fact_972_case__prodD,axiom,
    ! [Fa: fun_nat_fun_num_o,A2: nat,B2: num] :
      ( aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(Fa),aa_num5026351239867140595at_num(product_Pair_nat_num(A2),B2))
     => aa_num_o(aa_nat_fun_num_o(Fa,A2),B2) ) ).

% case_prodD
tff(fact_973_case__prodD,axiom,
    ! [Fa: fun_nat_fun_nat_o,A2: nat,B2: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(Fa),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2))
     => aa_nat_o(aa_nat_fun_nat_o(Fa,A2),B2) ) ).

% case_prodD
tff(fact_974_case__prodD,axiom,
    ! [Fa: fun_int_fun_int_o,A2: int,B2: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Fa),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2))
     => aa_int_o(aa_int_fun_int_o(Fa,A2),B2) ) ).

% case_prodD
tff(fact_975_case__prodE,axiom,
    ! [C: fun_num_fun_num_o,P2: product_prod_num_num] :
      ( aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(C),P2)
     => ~ ! [X2: num,Y2: num] :
            ( ( P2 = aa_num3156681643174609981um_num(product_Pair_num_num(X2),Y2) )
           => ~ aa_num_o(aa_num_fun_num_o(C,X2),Y2) ) ) ).

% case_prodE
tff(fact_976_case__prodE,axiom,
    ! [C: fun_na5540905182366533063VEBT_o,P2: produc8398139464844984134T_VEBT] :
      ( aa_Pro1497393596385541012VEBT_o(produc2834603712688810931VEBT_o(C),P2)
     => ~ ! [X2: nat,Y2: produc4813437837504472865T_VEBT] :
            ( ( P2 = aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(X2),Y2) )
           => ~ aa_Pro3461009536670225267VEBT_o(aa_nat6863640065291604488VEBT_o(C,X2),Y2) ) ) ).

% case_prodE
tff(fact_977_case__prodE,axiom,
    ! [C: fun_nat_fun_num_o,P2: product_prod_nat_num] :
      ( aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(C),P2)
     => ~ ! [X2: nat,Y2: num] :
            ( ( P2 = aa_num5026351239867140595at_num(product_Pair_nat_num(X2),Y2) )
           => ~ aa_num_o(aa_nat_fun_num_o(C,X2),Y2) ) ) ).

% case_prodE
tff(fact_978_case__prodE,axiom,
    ! [C: fun_nat_fun_nat_o,P2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2)
     => ~ ! [X2: nat,Y2: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y2) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(C,X2),Y2) ) ) ).

% case_prodE
tff(fact_979_case__prodE,axiom,
    ! [C: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2)
     => ~ ! [X2: int,Y2: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2) )
           => ~ aa_int_o(aa_int_fun_int_o(C,X2),Y2) ) ) ).

% case_prodE
tff(fact_980_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option2621746655072343315it_nat] :
      ( ( Option != none_P1551326421579882414it_nat )
    <=> aa_opt424737865643518209_nat_o(case_o535201446637900608it_nat($false,aTP_Lamm_fk),Option) ) ).

% option.disc_eq_case(2)
tff(fact_981_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option7339022715339332451it_nat] :
      ( ( Option != none_P7668321371905463026it_nat )
    <=> aa_opt4171472420520861111_nat_o(case_o1358941076187788256it_nat($false,aTP_Lamm_fl),Option) ) ).

% option.disc_eq_case(2)
tff(fact_982_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
    <=> aa_option_num_o(case_option_o_num($false,aTP_Lamm_fm),Option) ) ).

% option.disc_eq_case(2)
tff(fact_983_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
    <=> aa_option_nat_o(case_option_o_nat($false,aTP_Lamm_fn),Option) ) ).

% option.disc_eq_case(2)
tff(fact_984_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
    <=> aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat($false,aTP_Lamm_fo),Option) ) ).

% option.disc_eq_case(2)
tff(fact_985_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option2621746655072343315it_nat] :
      ( ( Option = none_P1551326421579882414it_nat )
    <=> aa_opt424737865643518209_nat_o(case_o535201446637900608it_nat($true,aTP_Lamm_fp),Option) ) ).

% option.disc_eq_case(1)
tff(fact_986_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option7339022715339332451it_nat] :
      ( ( Option = none_P7668321371905463026it_nat )
    <=> aa_opt4171472420520861111_nat_o(case_o1358941076187788256it_nat($true,aTP_Lamm_fq),Option) ) ).

% option.disc_eq_case(1)
tff(fact_987_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_num] :
      ( ( Option = none_num )
    <=> aa_option_num_o(case_option_o_num($true,aTP_Lamm_fr),Option) ) ).

% option.disc_eq_case(1)
tff(fact_988_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_nat] :
      ( ( Option = none_nat )
    <=> aa_option_nat_o(case_option_o_nat($true,aTP_Lamm_fs),Option) ) ).

% option.disc_eq_case(1)
tff(fact_989_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option = none_P5556105721700978146at_nat )
    <=> aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat($true,aTP_Lamm_ft),Option) ) ).

% option.disc_eq_case(1)
tff(fact_990_Product__Type_OCollect__case__prodD,axiom,
    ! [Xb: produc6575502325842934193n_assn,A3: fun_assn_fun_assn_o] :
      ( member7957490590177025114n_assn(Xb,collec1604292450182575004n_assn(produc7274209992780475162assn_o(A3)))
     => aa_assn_o(aa_assn_fun_assn_o(A3,aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Xb)),aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Xb)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_991_Product__Type_OCollect__case__prodD,axiom,
    ! [Xb: product_prod_nat_nat,A3: fun_nat_fun_nat_o] :
      ( member8440522571783428010at_nat(Xb,collec3392354462482085612at_nat(produc6081775807080527818_nat_o(A3)))
     => aa_nat_o(aa_nat_fun_nat_o(A3,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xb)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xb)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_992_Product__Type_OCollect__case__prodD,axiom,
    ! [Xb: product_prod_int_int,A3: fun_int_fun_int_o] :
      ( member5262025264175285858nt_int(Xb,collec213857154873943460nt_int(produc4947309494688390418_int_o(A3)))
     => aa_int_o(aa_int_fun_int_o(A3,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Xb)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_993_case__optionE,axiom,
    ! [P: $o,Q: fun_Pr7755520105644788814_nat_o,Xb: option2621746655072343315it_nat] :
      ( aa_opt424737865643518209_nat_o(case_o535201446637900608it_nat((P),Q),Xb)
     => ( ( ( Xb = none_P1551326421579882414it_nat )
         => ~ (P) )
       => ~ ! [Y2: produc120671012495760973it_nat] :
              ( ( Xb = some_P2407035485129114418it_nat(Y2) )
             => ~ aa_Pro2986953523225089223_nat_o(Q,Y2) ) ) ) ).

% case_optionE
tff(fact_994_case__optionE,axiom,
    ! [P: $o,Q: fun_Pr3989439477023515934_nat_o,Xb: option7339022715339332451it_nat] :
      ( aa_opt4171472420520861111_nat_o(case_o1358941076187788256it_nat((P),Q),Xb)
     => ( ( ( Xb = none_P7668321371905463026it_nat )
         => ~ (P) )
       => ~ ! [Y2: produc8047831477865546771it_nat] :
              ( ( Xb = some_P468703482102919278it_nat(Y2) )
             => ~ aa_Pro6395689520723908999_nat_o(Q,Y2) ) ) ) ).

% case_optionE
tff(fact_995_case__optionE,axiom,
    ! [P: $o,Q: fun_nat_o,Xb: option_nat] :
      ( aa_option_nat_o(case_option_o_nat((P),Q),Xb)
     => ( ( ( Xb = none_nat )
         => ~ (P) )
       => ~ ! [Y2: nat] :
              ( ( Xb = aa_nat_option_nat(some_nat,Y2) )
             => ~ aa_nat_o(Q,Y2) ) ) ) ).

% case_optionE
tff(fact_996_case__optionE,axiom,
    ! [P: $o,Q: fun_num_o,Xb: option_num] :
      ( aa_option_num_o(case_option_o_num((P),Q),Xb)
     => ( ( ( Xb = none_num )
         => ~ (P) )
       => ~ ! [Y2: num] :
              ( ( Xb = aa_num_option_num(some_num,Y2) )
             => ~ aa_num_o(Q,Y2) ) ) ) ).

% case_optionE
tff(fact_997_case__optionE,axiom,
    ! [P: $o,Q: fun_Pr4796943543331071514_nat_o,Xb: option4927543243414619207at_nat] :
      ( aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((P),Q),Xb)
     => ( ( ( Xb = none_P5556105721700978146at_nat )
         => ~ (P) )
       => ~ ! [Y2: product_prod_nat_nat] :
              ( ( Xb = some_P7363390416028606310at_nat(Y2) )
             => ~ aa_Pro6150637604445964947_nat_o(Q,Y2) ) ) ) ).

% case_optionE
tff(fact_998_less__eq__option__None,axiom,
    ! [Xb: option_num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,none_num),Xb) ).

% less_eq_option_None
tff(fact_999_less__eq__option__None,axiom,
    ! [Xb: option_nat] : aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,none_nat),Xb) ).

% less_eq_option_None
tff(fact_1000_less__eq__option__None__is__None,axiom,
    ! [Xb: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,Xb),none_num)
     => ( Xb = none_num ) ) ).

% less_eq_option_None_is_None
tff(fact_1001_less__eq__option__None__is__None,axiom,
    ! [Xb: option_nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,Xb),none_nat)
     => ( Xb = none_nat ) ) ).

% less_eq_option_None_is_None
tff(fact_1002_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_1003_less__option__None__Some,axiom,
    ! [Xb: nat] : aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,none_nat),aa_nat_option_nat(some_nat,Xb)) ).

% less_option_None_Some
tff(fact_1004_less__option__None__Some,axiom,
    ! [Xb: num] : aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,none_num),aa_num_option_num(some_num,Xb)) ).

% less_option_None_Some
tff(fact_1005_less__option__None__is__Some,axiom,
    ! [Xb: option_nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,none_nat),Xb)
     => ? [Z2: nat] : Xb = aa_nat_option_nat(some_nat,Z2) ) ).

% less_option_None_is_Some
tff(fact_1006_less__option__None__is__Some,axiom,
    ! [Xb: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_less_option_num,none_num),Xb)
     => ? [Z2: num] : Xb = aa_num_option_num(some_num,Z2) ) ).

% less_option_None_is_Some
tff(fact_1007_tdeletemimi_H,axiom,
    ! [Deg2: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Mi)),Deg2,TreeList),Summary2)),Xb)),one_one_nat) ) ).

% tdeletemimi'
tff(fact_1008_VEBT__internal_Ovebt__predi_H_Oraw__induct,axiom,
    ! [P: fun_VE3300235029744507244_nat_o,Xaa: produc3960855945107176009Ti_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: option_nat,N: nat] :
      ( ! [Vebt_predi: fun_VE3775134097089128360on_nat] :
          ( ! [A5: vEBT_VEBT,B4: vEBT_VEBTi,Ba: nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R2: option_nat,N4: nat] :
              ( aa_nat_o(heap_T306965388786959644on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(Vebt_predi,A5),B4),Ba),H3,H4,R2),N4)
             => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_nat3515811770666409442_nat_o(aa_VEB4657881624497257926_nat_o(aa_VEB1759150482313007205_nat_o(P,A5),B4),Ba),H3),H4),R2),N4) )
         => ! [T: vEBT_VEBT,Ti2: vEBT_VEBTi,X2: nat,Ta: heap_e7401611519738050253t_unit,Tia: heap_e7401611519738050253t_unit,Xa2: option_nat,N3: nat] :
              ( aa_nat_o(heap_T306965388786959644on_nat(aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_gi(Vebt_predi),T),X2),aTP_Lamm_bk(X2)),Ti2),Ta,Tia,Xa2),N3)
             => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_nat3515811770666409442_nat_o(aa_VEB4657881624497257926_nat_o(aa_VEB1759150482313007205_nat_o(P,T),Ti2),X2),Ta),Tia),Xa2),N3) ) )
     => ( aa_nat_o(heap_T306965388786959644on_nat(aa_Pro2727408364141080927on_nat(produc183673358652719894on_nat(produc3521483158534984758on_nat(vEBT_VEBT_vebt_predi)),Xaa),H,H2,R),N)
       => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(produc8196320604420998746_nat_o(produc1944727488097889274_nat_o(P),Xaa),H),H2),R),N) ) ) ).

% VEBT_internal.vebt_predi'.raw_induct
tff(fact_1009_VEBT__internal_Ovebt__succi_H_Oraw__induct,axiom,
    ! [P: fun_VE3300235029744507244_nat_o,Xaa: produc3960855945107176009Ti_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: option_nat,N: nat] :
      ( ! [Vebt_succi: fun_VE3775134097089128360on_nat] :
          ( ! [A5: vEBT_VEBT,B4: vEBT_VEBTi,Ba: nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R2: option_nat,N4: nat] :
              ( aa_nat_o(heap_T306965388786959644on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(Vebt_succi,A5),B4),Ba),H3,H4,R2),N4)
             => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_nat3515811770666409442_nat_o(aa_VEB4657881624497257926_nat_o(aa_VEB1759150482313007205_nat_o(P,A5),B4),Ba),H3),H4),R2),N4) )
         => ! [T: vEBT_VEBT,Ti2: vEBT_VEBTi,X2: nat,Ta: heap_e7401611519738050253t_unit,Tia: heap_e7401611519738050253t_unit,Xa2: option_nat,N3: nat] :
              ( aa_nat_o(heap_T306965388786959644on_nat(aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_gx(Vebt_succi),T),X2),aTP_Lamm_cq(X2)),Ti2),Ta,Tia,Xa2),N3)
             => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_nat3515811770666409442_nat_o(aa_VEB4657881624497257926_nat_o(aa_VEB1759150482313007205_nat_o(P,T),Ti2),X2),Ta),Tia),Xa2),N3) ) )
     => ( aa_nat_o(heap_T306965388786959644on_nat(aa_Pro2727408364141080927on_nat(produc183673358652719894on_nat(produc3521483158534984758on_nat(vEBT_VEBT_vebt_succi)),Xaa),H,H2,R),N)
       => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(produc8196320604420998746_nat_o(produc1944727488097889274_nat_o(P),Xaa),H),H2),R),N) ) ) ).

% VEBT_internal.vebt_succi'.raw_induct
tff(fact_1010_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
        & ( ( Xb = Mi )
          | ( Xb = Ma )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Ma)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ) ).

% member_inv
tff(fact_1011_insert__simp__mima,axiom,
    ! [Xb: nat,Mi: nat,Ma: nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( ( ( Xb = Mi )
        | ( Xb = Ma ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
       => ( vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2) ) ) ) ).

% insert_simp_mima
tff(fact_1012_vebt__predi_Oraw__induct,axiom,
    ! [P: fun_VE2372197504167591133_nat_o,Xaa: produc3881548065746020326Ti_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: option_nat,N: nat] :
      ( ! [Vebt_predi2: fun_VE8516060933855021849on_nat] :
          ( ! [A5: vEBT_VEBTi,B4: nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R2: option_nat,N4: nat] :
              ( aa_nat_o(heap_T306965388786959644on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(Vebt_predi2,A5),B4),H3,H4,R2),N4)
             => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_nat3515811770666409442_nat_o(aa_VEB4657881624497257926_nat_o(P,A5),B4),H3),H4),R2),N4) )
         => ! [T: vEBT_VEBTi,X2: nat,Ta: heap_e7401611519738050253t_unit,Xa2: heap_e7401611519738050253t_unit,R3: option_nat,N3: nat] :
              ( aa_nat_o(heap_T306965388786959644on_nat(aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_hd(Vebt_predi2),X2),aTP_Lamm_bk(X2)),T),Ta,Xa2,R3),N3)
             => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_nat3515811770666409442_nat_o(aa_VEB4657881624497257926_nat_o(P,T),X2),Ta),Xa2),R3),N3) ) )
     => ( aa_nat_o(heap_T306965388786959644on_nat(aa_Pro2606730266749126658on_nat(produc8911080112929139129on_nat(vEBT_vebt_predi),Xaa),H,H2,R),N)
       => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(produc4472639064241447549_nat_o(P,Xaa),H),H2),R),N) ) ) ).

% vebt_predi.raw_induct
tff(fact_1013_vebt__succi_Oraw__induct,axiom,
    ! [P: fun_VE2372197504167591133_nat_o,Xaa: produc3881548065746020326Ti_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: option_nat,N: nat] :
      ( ! [Vebt_succi2: fun_VE8516060933855021849on_nat] :
          ( ! [A5: vEBT_VEBTi,B4: nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R2: option_nat,N4: nat] :
              ( aa_nat_o(heap_T306965388786959644on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(Vebt_succi2,A5),B4),H3,H4,R2),N4)
             => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_nat3515811770666409442_nat_o(aa_VEB4657881624497257926_nat_o(P,A5),B4),H3),H4),R2),N4) )
         => ! [T: vEBT_VEBTi,X2: nat,Ta: heap_e7401611519738050253t_unit,Xa2: heap_e7401611519738050253t_unit,R3: option_nat,N3: nat] :
              ( aa_nat_o(heap_T306965388786959644on_nat(aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_hj(Vebt_succi2),X2),aTP_Lamm_cq(X2)),T),Ta,Xa2,R3),N3)
             => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(aa_nat3515811770666409442_nat_o(aa_VEB4657881624497257926_nat_o(P,T),X2),Ta),Xa2),R3),N3) ) )
     => ( aa_nat_o(heap_T306965388786959644on_nat(aa_Pro2606730266749126658on_nat(produc8911080112929139129on_nat(vEBT_vebt_succi),Xaa),H,H2,R),N)
       => aa_nat_o(aa_opt7444464167971770947_nat_o(aa_hea2436540527373857417_nat_o(aa_hea5443102814884448417_nat_o(produc4472639064241447549_nat_o(P,Xaa),H),H2),R),N) ) ) ).

% vebt_succi.raw_induct
tff(fact_1014_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xb: nat,Deg2: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Deg2)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),Xb) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_1015_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg2: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Deg2)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),Xb)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( Xb = Mi )
          | ( Xb = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_1016_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),Deg2)
     => ( ( Mi != Ma )
       => ( the_nat(vEBT_vebt_maxt(Summary2)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_1017_delt__out__of__range,axiom,
    ! [Xb: nat,Mi: nat,Ma: nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2) ) ) ) ).

% delt_out_of_range
tff(fact_1018_valid__tree__deg__neq__0,axiom,
    ! [Tb: vEBT_VEBT] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),zero_zero_nat) ).

% valid_tree_deg_neq_0
tff(fact_1019_valid__0__not,axiom,
    ! [Tb: vEBT_VEBT] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),zero_zero_nat) ).

% valid_0_not
tff(fact_1020_deg__deg__n,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary2)),N)
     => ( Deg2 = N ) ) ).

% deg_deg_n
tff(fact_1021_delete__pres__valid,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_vebt_delete(Tb,Xb)),N) ) ).

% delete_pres_valid
tff(fact_1022_deg__not__0,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% deg_not_0
tff(fact_1023_maxbmo,axiom,
    ! [Tb: vEBT_VEBT,Xb: nat] :
      ( ( vEBT_vebt_maxt(Tb) = aa_nat_option_nat(some_nat,Xb) )
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Tb),Xb) ) ).

% maxbmo
tff(fact_1024_dele__bmo__cont__corr,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,vEBT_vebt_delete(Tb,Xb)),Y)
      <=> ( ( Xb != Y )
          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Tb),Y) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_1025_both__member__options__equiv__member,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Tb),Xb)
      <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tb),Xb) ) ) ).

% both_member_options_equiv_member
tff(fact_1026_valid__member__both__member__options,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Tb),Xb)
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tb),Xb) ) ) ).

% valid_member_both_member_options
tff(fact_1027_dele__member__cont__corr,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_vebt_delete(Tb,Xb)),Y)
      <=> ( ( Xb != Y )
          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tb),Y) ) ) ) ).

% dele_member_cont_corr
tff(fact_1028_mint__member,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Maxi: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( vEBT_vebt_mint(Tb) = aa_nat_option_nat(some_nat,Maxi) )
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tb),Maxi) ) ) ).

% mint_member
tff(fact_1029_maxt__member,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Maxi: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( vEBT_vebt_maxt(Tb) = aa_nat_option_nat(some_nat,Maxi) )
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tb),Maxi) ) ) ).

% maxt_member
tff(fact_1030_mint__corr__help,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Mini: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( vEBT_vebt_mint(Tb) = aa_nat_option_nat(some_nat,Mini) )
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tb),Xb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mini),Xb) ) ) ) ).

% mint_corr_help
tff(fact_1031_maxt__corr__help,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Maxi: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( vEBT_vebt_maxt(Tb) = aa_nat_option_nat(some_nat,Maxi) )
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tb),Xb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_1032_split__part,axiom,
    ! [P: $o,Q: fun_nat_fun_nat_o,X3: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_hk((P)),Q)),X3)
    <=> ( (P)
        & aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(Q),X3) ) ) ).

% split_part
tff(fact_1033_split__part,axiom,
    ! [P: $o,Q: fun_int_fun_int_o,X3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_hl((P)),Q)),X3)
    <=> ( (P)
        & aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Q),X3) ) ) ).

% split_part
tff(fact_1034_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xb: nat,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tree),Xb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ) ).

% member_bound
tff(fact_1035_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Xb)
       => ( aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),Xb) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_1036_del__single__cont,axiom,
    ! [Xb: nat,Mi: nat,Ma: nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( ( ( Xb = Mi )
        & ( Xb = Ma ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary2) ) ) ) ).

% del_single_cont
tff(fact_1037_misiz,axiom,
    ! [Tb: vEBT_VEBT,N: nat,M: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( aa_nat_option_nat(some_nat,M) = vEBT_vebt_mint(Tb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ) ).

% misiz
tff(fact_1038_helpyd,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( aa_nat_option_nat(vEBT_vebt_succ(Tb),Xb) = aa_nat_option_nat(some_nat,Y) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ) ).

% helpyd
tff(fact_1039_helpypredd,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( aa_nat_option_nat(vEBT_vebt_pred(Tb),Xb) = aa_nat_option_nat(some_nat,Y) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ) ).

% helpypredd
tff(fact_1040_valid__insert__both__member__options__pres,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
         => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Tb),Xb)
           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(Tb,Y)),Xb) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_1041_valid__insert__both__member__options__add,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,vEBT_vebt_insert(Tb,Xb)),Xb) ) ) ).

% valid_insert_both_member_options_add
tff(fact_1042_post__member__pre__member,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
         => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_vebt_insert(Tb,Xb)),Y)
           => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tb),Y)
              | ( Xb = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_1043_i0__less,axiom,
    ! [N: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N)
    <=> ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_1044_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)) ) ) ).

% mi_ma_2_deg
tff(fact_1045_member__correct,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tb),Xb)
      <=> aa_set_nat_o(member_nat(Xb),vEBT_set_vebt(Tb)) ) ) ).

% member_correct
tff(fact_1046_both__member__options__ding,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary2)),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2))
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary2)),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_1047_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_1048_not__iless0,axiom,
    ! [N: extended_enat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(N),zero_z5237406670263579293d_enat) ).

% not_iless0
tff(fact_1049_enat__less__induct,axiom,
    ! [P: fun_Extended_enat_o,N: extended_enat] :
      ( ! [N3: extended_enat] :
          ( ! [M2: extended_enat] :
              ( aa_Extended_enat_o(ord_le72135733267957522d_enat(M2),N3)
             => aa_Extended_enat_o(P,M2) )
         => aa_Extended_enat_o(P,N3) )
     => aa_Extended_enat_o(P,N) ) ).

% enat_less_induct
tff(fact_1050_Collect__case__prod__mono,axiom,
    ! [A3: fun_nat_fun_nat_o,B5: fun_nat_fun_nat_o] :
      ( ord_le768339454706049738_nat_o(A3,B5)
     => ord_le3146513528884898305at_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(A3)),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(B5))) ) ).

% Collect_case_prod_mono
tff(fact_1051_Collect__case__prod__mono,axiom,
    ! [A3: fun_int_fun_int_o,B5: fun_int_fun_int_o] :
      ( ord_le77980682512674322_int_o(A3,B5)
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(produc4947309494688390418_int_o(A3)),collec213857154873943460nt_int(produc4947309494688390418_int_o(B5))) ) ).

% Collect_case_prod_mono
tff(fact_1052_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_nat_nat] : aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aTP_Lamm_hm),Prod) ).

% prod.disc_eq_case
tff(fact_1053_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_hn),Prod) ).

% prod.disc_eq_case
tff(fact_1054_in__children__def,axiom,
    ! [N: nat,TreeList: list_VEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(N,TreeList),Xb)
    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),N))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),N)) ) ).

% in_children_def
tff(fact_1055_maxt__corr,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( vEBT_vebt_maxt(Tb) = aa_nat_option_nat(some_nat,Xb) )
       => aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Tb)),Xb) ) ) ).

% maxt_corr
tff(fact_1056_maxt__sound,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Tb)),Xb)
       => ( vEBT_vebt_maxt(Tb) = aa_nat_option_nat(some_nat,Xb) ) ) ) ).

% maxt_sound
tff(fact_1057_mint__sound,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Tb)),Xb)
       => ( vEBT_vebt_mint(Tb) = aa_nat_option_nat(some_nat,Xb) ) ) ) ).

% mint_sound
tff(fact_1058_mint__corr,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( vEBT_vebt_mint(Tb) = aa_nat_option_nat(some_nat,Xb) )
       => aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Tb)),Xb) ) ) ).

% mint_corr
tff(fact_1059_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),N)
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Ma)
          & ? [M3: nat] :
              ( ( aa_nat_option_nat(some_nat,M3) = vEBT_vebt_mint(Summary2) )
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ).

% mintlistlength
tff(fact_1060_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,Summary2: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TreeList),Summary2)),Xb) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
tff(fact_1061_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X2),N) )
     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),N) ) ) ).

% set_n_deg_not_0
tff(fact_1062_set__vebt_H__def,axiom,
    ! [Tb: vEBT_VEBT] : vEBT_VEBT_set_vebt(Tb) = collect_nat(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tb)) ).

% set_vebt'_def
tff(fact_1063_vebt__pred_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xb: nat] :
      aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2)),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xb),
        aa_nat_option_nat(some_nat,Ma),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
              $let(
                minlow: option_nat,
                minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                $ite(
                  ( ( minlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h)),l)),
                  $let(
                    pr: option_nat,
                    pr:= aa_nat_option_nat(vEBT_vebt_pred(Summary2),h),
                    $ite(
                      pr = none_nat,
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xb),aa_nat_option_nat(some_nat,Mi),none_nat),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ),
              none_nat ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_1064_vebt__succ_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xb: nat] :
      aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2)),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi),
        aa_nat_option_nat(some_nat,Mi),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
              $let(
                maxlow: option_nat,
                maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                $ite(
                  ( ( maxlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_succ(nth_VEBT_VEBT(TreeList,h)),l)),
                  $let(
                    sc: option_nat,
                    sc:= aa_nat_option_nat(vEBT_vebt_succ(Summary2),h),
                    $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc))))) ) ) ),
              none_nat ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_1065_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),aa_nat_nat(suc,aa_nat_nat(suc,N)))
     => ? [Info: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] : Tree = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,aa_nat_nat(suc,N)),TreeList2),S2) ) ).

% deg_SUcn_Node
tff(fact_1066_set__vebt__set__vebt_H__valid,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( vEBT_set_vebt(Tb) = vEBT_VEBT_set_vebt(Tb) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_1067_inthall,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o,N: nat] :
      ( ! [X2: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(Xs))
         => aa_VEBT_VEBTi_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs))
       => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(Xs,N)) ) ) ).

% inthall
tff(fact_1068_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,N: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs))
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,N)) ) ) ).

% inthall
tff(fact_1069_inthall,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P: fun_Pr3571187171380813162assn_o,N: nat] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
         => aa_Pro2491721552408928739assn_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s6829681357464350627n_assn(Xs))
       => aa_Pro2491721552408928739assn_o(P,nth_Pr1769885009046257848n_assn(Xs,N)) ) ) ).

% inthall
tff(fact_1070_inthall,axiom,
    ! [Xs: list_real,P: fun_real_o,N: nat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => aa_real_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_real(Xs))
       => aa_real_o(P,aa_nat_real(nth_real(Xs),N)) ) ) ).

% inthall
tff(fact_1071_inthall,axiom,
    ! [Xs: list_o,P: fun_o_o,N: nat] :
      ( ! [X2: $o] :
          ( member_o((X2),set_o2(Xs))
         => aa_o_o(P,(X2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_o(Xs))
       => aa_o_o(P,aa_nat_o(nth_o(Xs),N)) ) ) ).

% inthall
tff(fact_1072_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_o,N: nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => aa_nat_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_nat(Xs))
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),N)) ) ) ).

% inthall
tff(fact_1073_inthall,axiom,
    ! [Xs: list_int,P: fun_int_o,N: nat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),set_int2(Xs))
         => aa_int_o(P,X2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_int(Xs))
       => aa_int_o(P,aa_nat_int(nth_int(Xs),N)) ) ) ).

% inthall
tff(fact_1074_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( aa_nat_nat(suc,X22) = aa_nat_nat(suc,Y22) )
    <=> ( X22 = Y22 ) ) ).

% nat.inject
tff(fact_1075_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( aa_nat_nat(suc,Nat) = aa_nat_nat(suc,Nat2) )
    <=> ( Nat = Nat2 ) ) ).

% old.nat.inject
tff(fact_1076_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),Deg2)
     => ( ( Mi = Ma )
       => ( ! [X3: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList))
             => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_13) )
          & ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X_13) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_1077_lessI,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(suc,N)) ).

% lessI
tff(fact_1078_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) ) ).

% Suc_mono
tff(fact_1079_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% Suc_less_eq
tff(fact_1080_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_nat_nat(suc,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M) ) ).

% Suc_le_mono
tff(fact_1081_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% diff_Suc_Suc
tff(fact_1082_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),K) ).

% Suc_diff_diff
tff(fact_1083_diff__0__eq__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_1084_diff__self__eq__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_1085_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_1086_diff__numeral__special_I9_J,axiom,
    minus_minus_uint32(one_one_uint32,one_one_uint32) = zero_zero_uint32 ).

% diff_numeral_special(9)
tff(fact_1087_diff__numeral__special_I9_J,axiom,
    minus_4019991460397169231l_num1(one_on7727431528512463931l_num1,one_on7727431528512463931l_num1) = zero_z3563351764282998399l_num1 ).

% diff_numeral_special(9)
tff(fact_1088_diff__numeral__special_I9_J,axiom,
    aa_real_real(minus_minus_real(one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_1089_diff__numeral__special_I9_J,axiom,
    minus_minus_rat(one_one_rat,one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_1090_diff__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_1091_power__0__Suc,axiom,
    ! [N: nat] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(zero_z3563351764282998399l_num1),aa_nat_nat(suc,N)) = zero_z3563351764282998399l_num1 ).

% power_0_Suc
tff(fact_1092_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,N)) = zero_zero_rat ).

% power_0_Suc
tff(fact_1093_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,N)) = zero_zero_nat ).

% power_0_Suc
tff(fact_1094_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,N)) = zero_zero_int ).

% power_0_Suc
tff(fact_1095_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,N)) = zero_zero_real ).

% power_0_Suc
tff(fact_1096_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),aa_nat_nat(suc,N)) = zero_zero_complex ).

% power_0_Suc
tff(fact_1097_power__0__Suc,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(zero_z3403309356797280102nteger),aa_nat_nat(suc,N)) = zero_z3403309356797280102nteger ).

% power_0_Suc
tff(fact_1098_power__Suc0__right,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_1099_power__Suc0__right,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_1100_power__Suc0__right,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_1101_power__Suc0__right,axiom,
    ! [A2: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_1102_power__Suc0__right,axiom,
    ! [A2: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_1103_less__Suc0,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(suc,zero_zero_nat))
    <=> ( N = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_1104_zero__less__Suc,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,N)) ).

% zero_less_Suc
tff(fact_1105_div__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,zero_zero_nat)) = M ).

% div_by_Suc_0
tff(fact_1106_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% zero_less_diff
tff(fact_1107_power__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_1108_nat__power__eq__Suc__0__iff,axiom,
    ! [Xb: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( Xb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_1109_diff__Suc__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),one_one_nat) = N ).

% diff_Suc_1
tff(fact_1110_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% diff_is_0_eq
tff(fact_1111_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_1112_Suc__pred,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% Suc_pred
tff(fact_1113_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% Suc_1
tff(fact_1114_div2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_1115_Suc__diff__1,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = N ) ) ).

% Suc_diff_1
tff(fact_1116_pred__member,axiom,
    ! [Tb: vEBT_VEBT,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Tb)),Xb),Y)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tb),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
        & ! [Z3: nat] :
            ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tb),Z3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),Xb) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z3),Y) ) ) ) ).

% pred_member
tff(fact_1117_succ__member,axiom,
    ! [Tb: vEBT_VEBT,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Tb)),Xb),Y)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tb),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        & ! [Z3: nat] :
            ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tb),Z3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z3) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z3) ) ) ) ).

% succ_member
tff(fact_1118_succ__corr,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Sx: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( aa_nat_option_nat(vEBT_vebt_succ(Tb),Xb) = aa_nat_option_nat(some_nat,Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Tb)),Xb),Sx) ) ) ).

% succ_corr
tff(fact_1119_pred__corr,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Px: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( aa_nat_option_nat(vEBT_vebt_pred(Tb),Xb) = aa_nat_option_nat(some_nat,Px) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Tb)),Xb),Px) ) ) ).

% pred_corr
tff(fact_1120_succ__correct,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Sx: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( aa_nat_option_nat(vEBT_vebt_succ(Tb),Xb) = aa_nat_option_nat(some_nat,Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_set_vebt(Tb)),Xb),Sx) ) ) ).

% succ_correct
tff(fact_1121_pred__correct,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Sx: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( aa_nat_option_nat(vEBT_vebt_pred(Tb),Xb) = aa_nat_option_nat(some_nat,Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_set_vebt(Tb)),Xb),Sx) ) ) ).

% pred_correct
tff(fact_1122_Suc__inject,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(suc,Xb) = aa_nat_nat(suc,Y) )
     => ( Xb = Y ) ) ).

% Suc_inject
tff(fact_1123_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa_nat_nat(suc,N) ).

% n_not_Suc_n
tff(fact_1124_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),K)),J) ).

% diff_commute
tff(fact_1125_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K: nat,I: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N3: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N3))
           => aa_nat_o(P,N3) )
       => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_1126_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(suc,M)) ).

% diff_less_Suc
tff(fact_1127_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ).

% Suc_diff_Suc
tff(fact_1128_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).

% Suc_diff_le
tff(fact_1129_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N) ).

% diff_Suc_eq_diff_pred
tff(fact_1130_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,I))),N) ) ).

% diff_Suc_less
tff(fact_1131_diff__divide__distrib,axiom,
    ! [A2: real,B2: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(A2),B2)),C) = aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A2),C)),aa_real_real(divide_divide_real(B2),C)) ).

% diff_divide_distrib
tff(fact_1132_diff__divide__distrib,axiom,
    ! [A2: rat,B2: rat,C: rat] : divide_divide_rat(minus_minus_rat(A2,B2),C) = minus_minus_rat(divide_divide_rat(A2,C),divide_divide_rat(B2,C)) ).

% diff_divide_distrib
tff(fact_1133_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).

% minus_nat.diff_0
tff(fact_1134_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M) = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
tff(fact_1135_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ) ).

% diff_less_mono2
tff(fact_1136_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),N)),K) ) ).

% less_imp_diff_less
tff(fact_1137_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K) )
        <=> ( M = N ) ) ) ) ).

% eq_diff_iff
tff(fact_1138_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ) ).

% le_diff_iff
tff(fact_1139_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1140_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),L)) ) ).

% diff_le_mono
tff(fact_1141_diff__le__self,axiom,
    ! [M: nat,N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M) ).

% diff_le_self
tff(fact_1142_le__diff__iff_H,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2) ) ) ) ).

% le_diff_iff'
tff(fact_1143_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ).

% diff_le_mono2
tff(fact_1144_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero_nat != aa_nat_nat(suc,X22) ).

% nat.distinct(1)
tff(fact_1145_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_1146_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_1147_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa_nat_nat(suc,X22) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_1148_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_1149_nat__induct,axiom,
    ! [P: fun_nat_o,N: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N3: nat] :
            ( aa_nat_o(P,N3)
           => aa_nat_o(P,aa_nat_nat(suc,N3)) )
       => aa_nat_o(P,N) ) ) ).

% nat_induct
tff(fact_1150_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M: nat,N: nat] :
      ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X2),zero_zero_nat)
     => ( ! [Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y2))
       => ( ! [X2: nat,Y2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X2),Y2)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X2)),aa_nat_nat(suc,Y2)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,M),N) ) ) ) ).

% diff_induct
tff(fact_1151_zero__induct,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N3: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N3))
           => aa_nat_o(P,N3) )
       => aa_nat_o(P,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_1152_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_1153_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_1154_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_1155_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M3: nat] : N = aa_nat_nat(suc,M3) ) ).

% not0_implies_Suc
tff(fact_1156_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_1157_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% Suc_lessD
tff(fact_1158_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_1159_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( ( aa_nat_nat(suc,M) != N )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),N) ) ) ).

% Suc_lessI
tff(fact_1160_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
       => ( M = N ) ) ) ).

% less_SucE
tff(fact_1161_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N)) ) ).

% less_SucI
tff(fact_1162_Ex__less__Suc,axiom,
    ! [N: nat,P: fun_nat_o] :
      ( ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,N))
          & aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,N)
        | ? [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),N)
            & aa_nat_o(P,I4) ) ) ) ).

% Ex_less_Suc
tff(fact_1163_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
        | ( M = N ) ) ) ).

% less_Suc_eq
tff(fact_1164_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(suc,M)) ) ).

% not_less_eq
tff(fact_1165_Nat_OAll__less__Suc,axiom,
    ! [N: nat,P: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,N))
         => aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,N)
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),N)
           => aa_nat_o(P,I4) ) ) ) ).

% Nat.All_less_Suc
tff(fact_1166_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M)
    <=> ? [M4: nat] :
          ( ( M = aa_nat_nat(suc,M4) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M4) ) ) ).

% Suc_less_eq2
tff(fact_1167_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(suc,M))
       => ( M = N ) ) ) ).

% less_antisym
tff(fact_1168_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% Suc_less_SucD
tff(fact_1169_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_1170_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I2),aa_nat_nat(suc,I2))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K2)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I2),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J2),K2)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I2),K2) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_1171_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I2: nat] :
            ( ( J = aa_nat_nat(suc,I2) )
           => aa_nat_o(P,I2) )
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
             => ( aa_nat_o(P,aa_nat_nat(suc,I2))
               => aa_nat_o(P,I2) ) )
         => aa_nat_o(P,I) ) ) ) ).

% strict_inc_induct
tff(fact_1172_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(suc,M))
      <=> ( N = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_1173_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% Suc_leD
tff(fact_1174_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_SucE
tff(fact_1175_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N)) ) ).

% le_SucI
tff(fact_1176_Suc__le__D,axiom,
    ! [N: nat,M5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N)),M5)
     => ? [M3: nat] : M5 = aa_nat_nat(suc,M3) ) ).

% Suc_le_D
tff(fact_1177_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
        | ( M = aa_nat_nat(suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_1178_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N)),N) ).

% Suc_n_not_le_n
tff(fact_1179_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N)),M) ) ).

% not_less_eq_eq
tff(fact_1180_full__nat__induct,axiom,
    ! [P: fun_nat_o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M2)),N3)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N3) )
     => aa_nat_o(P,N) ) ).

% full_nat_induct
tff(fact_1181_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(P,M)
       => ( ! [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N3)
             => ( aa_nat_o(P,N3)
               => aa_nat_o(P,aa_nat_nat(suc,N3)) ) )
         => aa_nat_o(P,N) ) ) ) ).

% nat_induct_at_least
tff(fact_1182_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R4: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( ! [X2: nat] : aa_nat_o(aa_nat_fun_nat_o(R4,X2),X2)
       => ( ! [X2: nat,Y2: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R4,X2),Y2)
             => ( aa_nat_o(aa_nat_fun_nat_o(R4,Y2),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(R4,X2),Z2) ) )
         => ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(R4,N3),aa_nat_nat(suc,N3))
           => aa_nat_o(aa_nat_fun_nat_o(R4,M),N) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_1183_Suc__pred_H,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_1184_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1185_div__if,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
        | ( N = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ).

% div_if
tff(fact_1186_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% div_geq
tff(fact_1187_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_1188_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),M) ) ) ).

% diff_less
tff(fact_1189_lift__Suc__mono__less,axiom,
    ! [Fa: fun_nat_real,N: nat,N5: nat] :
      ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(Fa,N3)),aa_nat_real(Fa,aa_nat_nat(suc,N3)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N5)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(Fa,N)),aa_nat_real(Fa,N5)) ) ) ).

% lift_Suc_mono_less
tff(fact_1190_lift__Suc__mono__less,axiom,
    ! [Fa: fun_nat_rat,N: nat,N5: nat] :
      ( ! [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,N3)),aa_nat_rat(Fa,aa_nat_nat(suc,N3)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N5)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,N)),aa_nat_rat(Fa,N5)) ) ) ).

% lift_Suc_mono_less
tff(fact_1191_lift__Suc__mono__less,axiom,
    ! [Fa: fun_nat_num,N: nat,N5: nat] :
      ( ! [N3: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(Fa,N3)),aa_nat_num(Fa,aa_nat_nat(suc,N3)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N5)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(Fa,N)),aa_nat_num(Fa,N5)) ) ) ).

% lift_Suc_mono_less
tff(fact_1192_lift__Suc__mono__less,axiom,
    ! [Fa: fun_nat_nat,N: nat,N5: nat] :
      ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(Fa,N3)),aa_nat_nat(Fa,aa_nat_nat(suc,N3)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N5)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(Fa,N)),aa_nat_nat(Fa,N5)) ) ) ).

% lift_Suc_mono_less
tff(fact_1193_lift__Suc__mono__less,axiom,
    ! [Fa: fun_nat_int,N: nat,N5: nat] :
      ( ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(Fa,N3)),aa_nat_int(Fa,aa_nat_nat(suc,N3)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N5)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(Fa,N)),aa_nat_int(Fa,N5)) ) ) ).

% lift_Suc_mono_less
tff(fact_1194_lift__Suc__mono__less__iff,axiom,
    ! [Fa: fun_nat_real,N: nat,M: nat] :
      ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(Fa,N3)),aa_nat_real(Fa,aa_nat_nat(suc,N3)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(Fa,N)),aa_nat_real(Fa,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1195_lift__Suc__mono__less__iff,axiom,
    ! [Fa: fun_nat_rat,N: nat,M: nat] :
      ( ! [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,N3)),aa_nat_rat(Fa,aa_nat_nat(suc,N3)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,N)),aa_nat_rat(Fa,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1196_lift__Suc__mono__less__iff,axiom,
    ! [Fa: fun_nat_num,N: nat,M: nat] :
      ( ! [N3: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(Fa,N3)),aa_nat_num(Fa,aa_nat_nat(suc,N3)))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(Fa,N)),aa_nat_num(Fa,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1197_lift__Suc__mono__less__iff,axiom,
    ! [Fa: fun_nat_nat,N: nat,M: nat] :
      ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(Fa,N3)),aa_nat_nat(Fa,aa_nat_nat(suc,N3)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(Fa,N)),aa_nat_nat(Fa,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1198_lift__Suc__mono__less__iff,axiom,
    ! [Fa: fun_nat_int,N: nat,M: nat] :
      ( ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(Fa,N3)),aa_nat_int(Fa,aa_nat_nat(suc,N3)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(Fa,N)),aa_nat_int(Fa,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1199_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ) ).

% less_diff_iff
tff(fact_1200_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),C)) ) ) ).

% diff_less_mono
tff(fact_1201_lift__Suc__mono__le,axiom,
    ! [Fa: fun_nat_set_nat,N: nat,N5: nat] :
      ( ! [N3: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(Fa,N3)),aa_nat_set_nat(Fa,aa_nat_nat(suc,N3)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(Fa,N)),aa_nat_set_nat(Fa,N5)) ) ) ).

% lift_Suc_mono_le
tff(fact_1202_lift__Suc__mono__le,axiom,
    ! [Fa: fun_nat_rat,N: nat,N5: nat] :
      ( ! [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,N3)),aa_nat_rat(Fa,aa_nat_nat(suc,N3)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,N)),aa_nat_rat(Fa,N5)) ) ) ).

% lift_Suc_mono_le
tff(fact_1203_lift__Suc__mono__le,axiom,
    ! [Fa: fun_nat_num,N: nat,N5: nat] :
      ( ! [N3: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(Fa,N3)),aa_nat_num(Fa,aa_nat_nat(suc,N3)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(Fa,N)),aa_nat_num(Fa,N5)) ) ) ).

% lift_Suc_mono_le
tff(fact_1204_lift__Suc__mono__le,axiom,
    ! [Fa: fun_nat_nat,N: nat,N5: nat] :
      ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Fa,N3)),aa_nat_nat(Fa,aa_nat_nat(suc,N3)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Fa,N)),aa_nat_nat(Fa,N5)) ) ) ).

% lift_Suc_mono_le
tff(fact_1205_lift__Suc__mono__le,axiom,
    ! [Fa: fun_nat_int,N: nat,N5: nat] :
      ( ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,N3)),aa_nat_int(Fa,aa_nat_nat(suc,N3)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,N)),aa_nat_int(Fa,N5)) ) ) ).

% lift_Suc_mono_le
tff(fact_1206_lift__Suc__antimono__le,axiom,
    ! [Fa: fun_nat_set_nat,N: nat,N5: nat] :
      ( ! [N3: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(Fa,aa_nat_nat(suc,N3))),aa_nat_set_nat(Fa,N3))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(Fa,N5)),aa_nat_set_nat(Fa,N)) ) ) ).

% lift_Suc_antimono_le
tff(fact_1207_lift__Suc__antimono__le,axiom,
    ! [Fa: fun_nat_rat,N: nat,N5: nat] :
      ( ! [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,aa_nat_nat(suc,N3))),aa_nat_rat(Fa,N3))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,N5)),aa_nat_rat(Fa,N)) ) ) ).

% lift_Suc_antimono_le
tff(fact_1208_lift__Suc__antimono__le,axiom,
    ! [Fa: fun_nat_num,N: nat,N5: nat] :
      ( ! [N3: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(Fa,aa_nat_nat(suc,N3))),aa_nat_num(Fa,N3))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(Fa,N5)),aa_nat_num(Fa,N)) ) ) ).

% lift_Suc_antimono_le
tff(fact_1209_lift__Suc__antimono__le,axiom,
    ! [Fa: fun_nat_nat,N: nat,N5: nat] :
      ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Fa,aa_nat_nat(suc,N3))),aa_nat_nat(Fa,N3))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Fa,N5)),aa_nat_nat(Fa,N)) ) ) ).

% lift_Suc_antimono_le
tff(fact_1210_lift__Suc__antimono__le,axiom,
    ! [Fa: fun_nat_int,N: nat,N5: nat] :
      ( ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,aa_nat_nat(suc,N3))),aa_nat_int(Fa,N3))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N5)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,N5)),aa_nat_int(Fa,N)) ) ) ).

% lift_Suc_antimono_le
tff(fact_1211_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_o] :
      ( ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,N))
          & aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        | ? [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),N)
            & aa_nat_o(P,aa_nat_nat(suc,I4)) ) ) ) ).

% Ex_less_Suc2
tff(fact_1212_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
    <=> ? [M6: nat] : N = aa_nat_nat(suc,M6) ) ).

% gr0_conv_Suc
tff(fact_1213_All__less__Suc2,axiom,
    ! [N: nat,P: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,N))
         => aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),N)
           => aa_nat_o(P,aa_nat_nat(suc,I4)) ) ) ) ).

% All_less_Suc2
tff(fact_1214_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ? [M3: nat] : N = aa_nat_nat(suc,M3) ) ).

% gr0_implies_Suc
tff(fact_1215_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N))
    <=> ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),N) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_1216_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_1217_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N) ) ).

% Suc_leI
tff(fact_1218_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% Suc_le_eq
tff(fact_1219_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,I)
       => ( ! [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),J)
               => ( aa_nat_o(P,N3)
                 => aa_nat_o(P,aa_nat_nat(suc,N3)) ) ) )
         => aa_nat_o(P,J) ) ) ) ).

% dec_induct
tff(fact_1220_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,J)
       => ( ! [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),J)
               => ( aa_nat_o(P,aa_nat_nat(suc,N3))
                 => aa_nat_o(P,N3) ) ) )
         => aa_nat_o(P,I) ) ) ) ).

% inc_induct
tff(fact_1221_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% Suc_le_lessD
tff(fact_1222_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(suc,M))
      <=> ( N = M ) ) ) ).

% le_less_Suc_eq
tff(fact_1223_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% less_Suc_eq_le
tff(fact_1224_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N)),M) ) ).

% less_eq_Suc_le
tff(fact_1225_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,N)) ) ).

% le_imp_less_Suc
tff(fact_1226_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),N)) ).

% Suc_div_le_mono
tff(fact_1227_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(K),M)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_1228_power__inject__base,axiom,
    ! [A2: real,N: nat,B2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_nat_nat(suc,N)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
tff(fact_1229_power__inject__base,axiom,
    ! [A2: code_integer,N: nat,B2: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N)) = aa_nat_Code_integer(power_8256067586552552935nteger(B2),aa_nat_nat(suc,N)) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
tff(fact_1230_power__inject__base,axiom,
    ! [A2: rat,N: nat,B2: rat] :
      ( ( aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N)) = aa_nat_rat(power_power_rat(B2),aa_nat_nat(suc,N)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
tff(fact_1231_power__inject__base,axiom,
    ! [A2: nat,N: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(suc,N)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
tff(fact_1232_power__inject__base,axiom,
    ! [A2: int,N: nat,B2: int] :
      ( ( aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N)) = aa_nat_int(power_power_int(B2),aa_nat_nat(suc,N)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
tff(fact_1233_power__le__imp__le__base,axiom,
    ! [A2: real,N: nat,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_nat_nat(suc,N)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ).

% power_le_imp_le_base
tff(fact_1234_power__le__imp__le__base,axiom,
    ! [A2: code_integer,N: nat,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N))),aa_nat_Code_integer(power_8256067586552552935nteger(B2),aa_nat_nat(suc,N)))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2) ) ) ).

% power_le_imp_le_base
tff(fact_1235_power__le__imp__le__base,axiom,
    ! [A2: rat,N: nat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N))),aa_nat_rat(power_power_rat(B2),aa_nat_nat(suc,N)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ) ).

% power_le_imp_le_base
tff(fact_1236_power__le__imp__le__base,axiom,
    ! [A2: nat,N: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ) ).

% power_le_imp_le_base
tff(fact_1237_power__le__imp__le__base,axiom,
    ! [A2: int,N: nat,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N))),aa_nat_int(power_power_int(B2),aa_nat_nat(suc,N)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ) ).

% power_le_imp_le_base
tff(fact_1238_power__gt1,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N))) ) ).

% power_gt1
tff(fact_1239_power__gt1,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N))) ) ).

% power_gt1
tff(fact_1240_power__gt1,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N))) ) ).

% power_gt1
tff(fact_1241_power__gt1,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N))) ) ).

% power_gt1
tff(fact_1242_power__gt1,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N))) ) ).

% power_gt1
tff(fact_1243_numeral__1__eq__Suc__0,axiom,
    aa_num_nat(numeral_numeral_nat,one) = aa_nat_nat(suc,zero_zero_nat) ).

% numeral_1_eq_Suc_0
tff(fact_1244_ex__least__nat__less,axiom,
    ! [P: fun_nat_o,N: nat] :
      ( aa_nat_o(P,N)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),N)
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),K2)
               => ~ aa_nat_o(P,I3) )
            & aa_nat_o(P,aa_nat_nat(suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_1245_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3)
             => ( aa_nat_o(P,N3)
               => aa_nat_o(P,aa_nat_nat(suc,N3)) ) )
         => aa_nat_o(P,N) ) ) ) ).

% nat_induct_non_zero
tff(fact_1246_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),K)) ) ).

% power_gt_expt
tff(fact_1247_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N)) ) ).

% nat_one_le_power
tff(fact_1248_power2__commute,axiom,
    ! [Xb: complex,Y: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_1249_power2__commute,axiom,
    ! [Xb: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_1250_power2__commute,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_1251_power2__commute,axiom,
    ! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(Xb,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(minus_minus_rat(Y,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_1252_power2__commute,axiom,
    ! [Xb: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_1253_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)),Vi) = none_nat ).

% vebt_succ.simps(5)
tff(fact_1254_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)),Vj) = none_nat ).

% vebt_pred.simps(6)
tff(fact_1255_power__diff,axiom,
    ! [A2: complex,N: nat,M: nat] :
      ( ( A2 != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),M)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)) ) ) ) ).

% power_diff
tff(fact_1256_power__diff,axiom,
    ! [A2: code_integer,N: nat,M: nat] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ) ) ) ).

% power_diff
tff(fact_1257_power__diff,axiom,
    ! [A2: real,N: nat,M: nat] :
      ( ( A2 != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ) ) ) ).

% power_diff
tff(fact_1258_power__diff,axiom,
    ! [A2: rat,N: nat,M: nat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ( aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = divide_divide_rat(aa_nat_rat(power_power_rat(A2),M),aa_nat_rat(power_power_rat(A2),N)) ) ) ) ).

% power_diff
tff(fact_1259_power__diff,axiom,
    ! [A2: nat,N: nat,M: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ) ) ) ).

% power_diff
tff(fact_1260_power__diff,axiom,
    ! [A2: int,N: nat,M: nat] :
      ( ( A2 != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ( aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),N)) ) ) ) ).

% power_diff
tff(fact_1261_power__Suc__le__self,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N))),A2) ) ) ).

% power_Suc_le_self
tff(fact_1262_power__Suc__le__self,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N))),A2) ) ) ).

% power_Suc_le_self
tff(fact_1263_power__Suc__le__self,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N))),A2) ) ) ).

% power_Suc_le_self
tff(fact_1264_power__Suc__le__self,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N))),A2) ) ) ).

% power_Suc_le_self
tff(fact_1265_power__Suc__le__self,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N))),A2) ) ) ).

% power_Suc_le_self
tff(fact_1266_power__Suc__less__one,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N))),one_one_Code_integer) ) ) ).

% power_Suc_less_one
tff(fact_1267_power__Suc__less__one,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_1268_power__Suc__less__one,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_1269_power__Suc__less__one,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_1270_power__Suc__less__one,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_1271_option_Osize_I4_J,axiom,
    ! [X22: nat] : size_size_option_nat(aa_nat_option_nat(some_nat,X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_1272_option_Osize_I4_J,axiom,
    ! [X22: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_1273_option_Osize_I4_J,axiom,
    ! [X22: num] : size_size_option_num(aa_num_option_num(some_num,X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_1274_option_Osize_I3_J,axiom,
    size_s3991424295186984831it_nat(none_P1551326421579882414it_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1275_option_Osize_I3_J,axiom,
    size_s364044314319911927it_nat(none_P7668321371905463026it_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1276_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1277_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1278_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1279_numeral__2__eq__2,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% numeral_2_eq_2
tff(fact_1280_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,Summary2: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary2)),Xb) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
tff(fact_1281_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_1282_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N) != zero_z3563351764282998399l_num1 )
     => ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_z3563351764282998399l_num1 ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_1283_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_1284_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_1285_power__diff__power__eq,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% power_diff_power_eq
tff(fact_1286_power__diff__power__eq,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% power_diff_power_eq
tff(fact_1287_power__diff__power__eq,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M),aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ).

% power_diff_power_eq
tff(fact_1288_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N))) ) ).

% diff_le_diff_pow
tff(fact_1289_less__2__cases,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_1290_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( N = zero_zero_nat )
        | ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_1291_power__sub,axiom,
    ! [N: nat,M: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ) ) ) ).

% power_sub
tff(fact_1292_power__minus__is__div,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)) ) ) ).

% power_minus_is_div
tff(fact_1293_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_1294_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_1295_less__two__pow__divD,axiom,
    ! [Xb: nat,N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% less_two_pow_divD
tff(fact_1296_less__two__pow__divI,axiom,
    ! [Xb: nat,N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ) ).

% less_two_pow_divI
tff(fact_1297_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] : aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)),Va) = none_nat ).

% vebt_succ.simps(3)
tff(fact_1298_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] : aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va)),Vb) = none_nat ).

% vebt_pred.simps(4)
tff(fact_1299_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,
    ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary2)),Uu) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
tff(fact_1300_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : aa_nat_option_nat(vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc),Vd)),Ve) = none_nat ).

% vebt_succ.simps(4)
tff(fact_1301_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : aa_nat_option_nat(vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd),Ve)),Vf) = none_nat ).

% vebt_pred.simps(5)
tff(fact_1302_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2)),Xb)
    <=> $ite(
          Xb = Mi,
          $true,
          $ite(
            Xb = Ma,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xb),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_1303_Suc__diff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_diff
tff(fact_1304_not__Some__eq2,axiom,
    ! [V: option2661157926820139483um_num] :
      ( ! [X: num,Y3: num] : V != some_P6201964756284913402um_num(aa_num3156681643174609981um_num(product_Pair_num_num(X),Y3))
    <=> ( V = none_P4394680061957285238um_num ) ) ).

% not_Some_eq2
tff(fact_1305_not__Some__eq2,axiom,
    ! [V: option254855292876462358T_VEBT] :
      ( ! [X: nat,Y3: produc4813437837504472865T_VEBT] : V != some_P5996745733903548321T_VEBT(aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(X),Y3))
    <=> ( V = none_P330983522480640549T_VEBT ) ) ).

% not_Some_eq2
tff(fact_1306_not__Some__eq2,axiom,
    ! [V: option642762832853965969at_num] :
      ( ! [X: nat,Y3: num] : V != some_P8071634352977444016at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(X),Y3))
    <=> ( V = none_P6264349658649815852at_num ) ) ).

% not_Some_eq2
tff(fact_1307_not__Some__eq2,axiom,
    ! [V: option4624381673175914239nt_int] :
      ( ! [X: int,Y3: int] : V != some_P4184893108420464158nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y3))
    <=> ( V = none_P2377608414092835994nt_int ) ) ).

% not_Some_eq2
tff(fact_1308_not__Some__eq2,axiom,
    ! [V: option2621746655072343315it_nat] :
      ( ! [X: option_nat,Y3: produc6653097349344004940it_nat] : V != some_P2407035485129114418it_nat(produc61566615109097733it_nat(X,Y3))
    <=> ( V = none_P1551326421579882414it_nat ) ) ).

% not_Some_eq2
tff(fact_1309_not__Some__eq2,axiom,
    ! [V: option7339022715339332451it_nat] :
      ( ! [X: $o,Y3: produc6653097349344004940it_nat] : V != some_P468703482102919278it_nat(produc6655106138504972685it_nat((X),Y3))
    <=> ( V = none_P7668321371905463026it_nat ) ) ).

% not_Some_eq2
tff(fact_1310_not__Some__eq2,axiom,
    ! [V: option4927543243414619207at_nat] :
      ( ! [X: nat,Y3: nat] : V != some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y3))
    <=> ( V = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq2
tff(fact_1311_vebt__pred_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( aa_nat_option_nat(vEBT_vebt_pred(Xb),Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv: $o] : Xb = vEBT_Leaf((Uu2),(Uv))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A: $o] :
              ( ? [Uw: $o] : Xb = vEBT_Leaf((A),(Uw))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != $ite((A),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) ) ) )
         => ( ! [A: $o,B: $o] :
                ( ( Xb = vEBT_Leaf((A),(B)) )
               => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
                 => ( Y != $ite(
                        (B),
                        aa_nat_option_nat(some_nat,one_one_nat),
                        $ite((A),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(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] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( Y != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                              aa_nat_option_nat(some_nat,Ma2),
                              $let(
                                l: nat,
                                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                    $let(
                                      minlow: option_nat,
                                      minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                      $ite(
                                        ( ( minlow != none_nat )
                                        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                                        aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,h)),l)),
                                        $let(
                                          pr: option_nat,
                                          pr:= aa_nat_option_nat(vEBT_vebt_pred(Summary),h),
                                          $ite(
                                            pr = none_nat,
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Xaa),aa_nat_option_nat(some_nat,Mi2),none_nat),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat(pr)))) ) ) ) ),
                                    none_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_1312_vebt__succ_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( aa_nat_option_nat(vEBT_vebt_succ(Xb),Xaa) = Y )
     => ( ! [Uu2: $o,B: $o] :
            ( ( Xb = vEBT_Leaf((Uu2),(B)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y != $ite((B),aa_nat_option_nat(some_nat,one_one_nat),none_nat) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : Xb = vEBT_Leaf((Uv),(Uw))
           => ( ? [N3: nat] : Xaa = aa_nat_nat(suc,N3)
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(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] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( Y != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                            aa_nat_option_nat(some_nat,Mi2),
                            $let(
                              l: nat,
                              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                  $let(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,h)),l)),
                                      $let(
                                        sc: option_nat,
                                        sc:= aa_nat_option_nat(vEBT_vebt_succ(Summary),h),
                                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(sc))))) ) ) ),
                                  none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_1313_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(minus_minus_real(A2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ).

% zero_comp_diff_simps(2)
tff(fact_1314_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),minus_minus_rat(A2,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ).

% zero_comp_diff_simps(2)
tff(fact_1315_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2) ) ).

% zero_comp_diff_simps(2)
tff(fact_1316_diff__gt__0__iff__gt,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(minus_minus_real(A2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ).

% diff_gt_0_iff_gt
tff(fact_1317_diff__gt__0__iff__gt,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),minus_minus_rat(A2,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ).

% diff_gt_0_iff_gt
tff(fact_1318_diff__gt__0__iff__gt,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2) ) ).

% diff_gt_0_iff_gt
tff(fact_1319_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(A2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ).

% zero_comp_diff_simps(1)
tff(fact_1320_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),minus_minus_rat(A2,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ).

% zero_comp_diff_simps(1)
tff(fact_1321_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ).

% zero_comp_diff_simps(1)
tff(fact_1322_diff__ge__0__iff__ge,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(A2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ).

% diff_ge_0_iff_ge
tff(fact_1323_diff__ge__0__iff__ge,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),minus_minus_rat(A2,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ).

% diff_ge_0_iff_ge
tff(fact_1324_diff__ge__0__iff__ge,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ).

% diff_ge_0_iff_ge
tff(fact_1325_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary2),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xb)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary2) ).

% vebt_insert.simps(4)
tff(fact_1326_inrange,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_set_nat_o(ord_less_eq_set_nat(vEBT_VEBT_set_vebt(Tb)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))) ) ).

% inrange
tff(fact_1327_Leaf__0__not,axiom,
    ! [A2: $o,B2: $o] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Leaf((A2),(B2))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_1328_deg__1__Leafy,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( N = one_one_nat )
       => ? [A: $o,B: $o] : Tb = vEBT_Leaf((A),(B)) ) ) ).

% deg_1_Leafy
tff(fact_1329_deg__1__Leaf,axiom,
    ! [Tb: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),one_one_nat)
     => ? [A: $o,B: $o] : Tb = vEBT_Leaf((A),(B)) ) ).

% deg_1_Leaf
tff(fact_1330_deg1Leaf,axiom,
    ! [Tb: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),one_one_nat)
    <=> ? [A6: $o,B6: $o] : Tb = vEBT_Leaf((A6),(B6)) ) ).

% deg1Leaf
tff(fact_1331_idiff__0__right,axiom,
    ! [N: extended_enat] : minus_3235023915231533773d_enat(N,zero_z5237406670263579293d_enat) = N ).

% idiff_0_right
tff(fact_1332_idiff__0,axiom,
    ! [N: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,N) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_1333_le__zero__eq,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),zero_zero_nat)
    <=> ( N = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_1334_not__gr__zero,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
    <=> ( N = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_1335_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: word_N3645301735248828278l_num1] : minus_4019991460397169231l_num1(A2,A2) = zero_z3563351764282998399l_num1 ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1336_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: real] : aa_real_real(minus_minus_real(A2),A2) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1337_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: rat] : minus_minus_rat(A2,A2) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1338_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),A2) = zero_zero_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1339_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),A2) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1340_diff__zero,axiom,
    ! [A2: word_N3645301735248828278l_num1] : minus_4019991460397169231l_num1(A2,zero_z3563351764282998399l_num1) = A2 ).

% diff_zero
tff(fact_1341_diff__zero,axiom,
    ! [A2: real] : aa_real_real(minus_minus_real(A2),zero_zero_real) = A2 ).

% diff_zero
tff(fact_1342_diff__zero,axiom,
    ! [A2: rat] : minus_minus_rat(A2,zero_zero_rat) = A2 ).

% diff_zero
tff(fact_1343_diff__zero,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),zero_zero_nat) = A2 ).

% diff_zero
tff(fact_1344_diff__zero,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),zero_zero_int) = A2 ).

% diff_zero
tff(fact_1345_zero__diff,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A2) = zero_zero_nat ).

% zero_diff
tff(fact_1346_diff__0__right,axiom,
    ! [A2: word_N3645301735248828278l_num1] : minus_4019991460397169231l_num1(A2,zero_z3563351764282998399l_num1) = A2 ).

% diff_0_right
tff(fact_1347_diff__0__right,axiom,
    ! [A2: real] : aa_real_real(minus_minus_real(A2),zero_zero_real) = A2 ).

% diff_0_right
tff(fact_1348_diff__0__right,axiom,
    ! [A2: rat] : minus_minus_rat(A2,zero_zero_rat) = A2 ).

% diff_0_right
tff(fact_1349_diff__0__right,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),zero_zero_int) = A2 ).

% diff_0_right
tff(fact_1350_diff__self,axiom,
    ! [A2: word_N3645301735248828278l_num1] : minus_4019991460397169231l_num1(A2,A2) = zero_z3563351764282998399l_num1 ).

% diff_self
tff(fact_1351_diff__self,axiom,
    ! [A2: real] : aa_real_real(minus_minus_real(A2),A2) = zero_zero_real ).

% diff_self
tff(fact_1352_diff__self,axiom,
    ! [A2: rat] : minus_minus_rat(A2,A2) = zero_zero_rat ).

% diff_self
tff(fact_1353_diff__self,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),A2) = zero_zero_int ).

% diff_self
tff(fact_1354_vebt__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,vEBT_Leaf((A2),(B2))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (A2),
          $ite(Xb = one_one_nat,(B2),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_1355_vebt__insert_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A2),(B2)),Xb) = $ite(
        Xb = zero_zero_nat,
        vEBT_Leaf($true,(B2)),
        $ite(Xb = one_one_nat,vEBT_Leaf((A2),$true),vEBT_Leaf((A2),(B2))) ) ).

% vebt_insert.simps(1)
tff(fact_1356_zero__reorient,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( ( zero_z3563351764282998399l_num1 = Xb )
    <=> ( Xb = zero_z3563351764282998399l_num1 ) ) ).

% zero_reorient
tff(fact_1357_zero__reorient,axiom,
    ! [Xb: real] :
      ( ( zero_zero_real = Xb )
    <=> ( Xb = zero_zero_real ) ) ).

% zero_reorient
tff(fact_1358_zero__reorient,axiom,
    ! [Xb: rat] :
      ( ( zero_zero_rat = Xb )
    <=> ( Xb = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_1359_zero__reorient,axiom,
    ! [Xb: nat] :
      ( ( zero_zero_nat = Xb )
    <=> ( Xb = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_1360_zero__reorient,axiom,
    ! [Xb: int] :
      ( ( zero_zero_int = Xb )
    <=> ( Xb = zero_zero_int ) ) ).

% zero_reorient
tff(fact_1361_ord__eq__le__eq__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
      ( ( A2 = B2 )
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C)
       => ( ( C = D )
         => aa_set_nat_o(ord_less_eq_set_nat(A2),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1362_ord__eq__le__eq__trans,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( A2 = B2 )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C)
       => ( ( C = D )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1363_ord__eq__le__eq__trans,axiom,
    ! [A2: num,B2: num,C: num,D: num] :
      ( ( A2 = B2 )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C)
       => ( ( C = D )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1364_ord__eq__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( ( A2 = B2 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C)
       => ( ( C = D )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1365_ord__eq__le__eq__trans,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( A2 = B2 )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C)
       => ( ( C = D )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),D) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_1366_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,N: nat] : aa_nat_option_nat(vEBT_vebt_succ(vEBT_Leaf((Uv2),(Uw2))),aa_nat_nat(suc,N)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_1367_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o] : aa_nat_option_nat(vEBT_vebt_pred(vEBT_Leaf((Uu),(Uv2))),zero_zero_nat) = none_nat ).

% vebt_pred.simps(1)
tff(fact_1368_one__reorient,axiom,
    ! [Xb: uint32] :
      ( ( one_one_uint32 = Xb )
    <=> ( Xb = one_one_uint32 ) ) ).

% one_reorient
tff(fact_1369_one__reorient,axiom,
    ! [Xb: real] :
      ( ( one_one_real = Xb )
    <=> ( Xb = one_one_real ) ) ).

% one_reorient
tff(fact_1370_one__reorient,axiom,
    ! [Xb: rat] :
      ( ( one_one_rat = Xb )
    <=> ( Xb = one_one_rat ) ) ).

% one_reorient
tff(fact_1371_one__reorient,axiom,
    ! [Xb: nat] :
      ( ( one_one_nat = Xb )
    <=> ( Xb = one_one_nat ) ) ).

% one_reorient
tff(fact_1372_one__reorient,axiom,
    ! [Xb: int] :
      ( ( one_one_int = Xb )
    <=> ( Xb = one_one_int ) ) ).

% one_reorient
tff(fact_1373_bex2I,axiom,
    ! [A2: num,B2: num,S3: set_Pr8218934625190621173um_num,P: fun_num_fun_num_o] :
      ( member7279096912039735102um_num(aa_num3156681643174609981um_num(product_Pair_num_num(A2),B2),S3)
     => ( ( member7279096912039735102um_num(aa_num3156681643174609981um_num(product_Pair_num_num(A2),B2),S3)
         => aa_num_o(aa_num_fun_num_o(P,A2),B2) )
       => ? [A: num,B: num] :
            ( member7279096912039735102um_num(aa_num3156681643174609981um_num(product_Pair_num_num(A),B),S3)
            & aa_num_o(aa_num_fun_num_o(P,A),B) ) ) ) ).

% bex2I
tff(fact_1374_bex2I,axiom,
    ! [A2: nat,B2: produc4813437837504472865T_VEBT,S3: set_Pr563407847431865468T_VEBT,P: fun_na5540905182366533063VEBT_o] :
      ( member306291179834725981T_VEBT(aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A2),B2),S3)
     => ( ( member306291179834725981T_VEBT(aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A2),B2),S3)
         => aa_Pro3461009536670225267VEBT_o(aa_nat6863640065291604488VEBT_o(P,A2),B2) )
       => ? [A: nat,B: produc4813437837504472865T_VEBT] :
            ( member306291179834725981T_VEBT(aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A),B),S3)
            & aa_Pro3461009536670225267VEBT_o(aa_nat6863640065291604488VEBT_o(P,A),B) ) ) ) ).

% bex2I
tff(fact_1375_bex2I,axiom,
    ! [A2: nat,B2: num,S3: set_Pr6200539531224447659at_num,P: fun_nat_fun_num_o] :
      ( member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(A2),B2),S3)
     => ( ( member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(A2),B2),S3)
         => aa_num_o(aa_nat_fun_num_o(P,A2),B2) )
       => ? [A: nat,B: num] :
            ( member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(A),B),S3)
            & aa_num_o(aa_nat_fun_num_o(P,A),B) ) ) ) ).

% bex2I
tff(fact_1376_bex2I,axiom,
    ! [A2: nat,B2: nat,S3: set_Pr1261947904930325089at_nat,P: fun_nat_fun_nat_o] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2),S3)
     => ( ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2),S3)
         => aa_nat_o(aa_nat_fun_nat_o(P,A2),B2) )
       => ? [A: nat,B: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B),S3)
            & aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% bex2I
tff(fact_1377_bex2I,axiom,
    ! [A2: int,B2: int,S3: set_Pr958786334691620121nt_int,P: fun_int_fun_int_o] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2),S3)
     => ( ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2),S3)
         => aa_int_o(aa_int_fun_int_o(P,A2),B2) )
       => ? [A: int,B: int] :
            ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B),S3)
            & aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ) ).

% bex2I
tff(fact_1378_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,
    ! [A2: $o,B2: $o,N: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,vEBT_Leaf((A2),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,N))) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
tff(fact_1379_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,
    ! [A2: $o,B2: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,vEBT_Leaf((A2),(B2))),zero_zero_nat) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
tff(fact_1380_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,
    ! [A2: $o,B2: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,vEBT_Leaf((A2),(B2))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
tff(fact_1381_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),Xb),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        & ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),X)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_1382_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),Xb),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
        & ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Xb)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_1383_vebt__pred_Osimps_I2_J,axiom,
    ! [A2: $o,Uw2: $o] :
      aa_nat_option_nat(vEBT_vebt_pred(vEBT_Leaf((A2),(Uw2))),aa_nat_nat(suc,zero_zero_nat)) = $ite((A2),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) ).

% vebt_pred.simps(2)
tff(fact_1384_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] :
      aa_nat_option_nat(vEBT_vebt_succ(vEBT_Leaf((Uu),(B2))),zero_zero_nat) = $ite((B2),aa_nat_option_nat(some_nat,one_one_nat),none_nat) ).

% vebt_succ.simps(1)
tff(fact_1385_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,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A: $o,B: $o] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),zero_zero_nat)
     => ( ! [A: $o,B: $o] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A: $o,B: $o,N3: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,N3)))
         => ( ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2),Summary)),Uu2)
           => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,TreeList2),Summary)),X2)
             => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary)),X2)
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),X2) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
tff(fact_1386_vebt__pred_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Va: nat] :
      aa_nat_option_nat(vEBT_vebt_pred(vEBT_Leaf((A2),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
        (B2),
        aa_nat_option_nat(some_nat,one_one_nat),
        $ite((A2),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) ) ).

% vebt_pred.simps(3)
tff(fact_1387_zero__le,axiom,
    ! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb) ).

% zero_le
tff(fact_1388_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
    <=> ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_1389_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_1390_not__less__zero,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),zero_zero_nat) ).

% not_less_zero
tff(fact_1391_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% gr_zeroI
tff(fact_1392_eq__iff__diff__eq__0,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( A2 = B2 )
    <=> ( minus_4019991460397169231l_num1(A2,B2) = zero_z3563351764282998399l_num1 ) ) ).

% eq_iff_diff_eq_0
tff(fact_1393_eq__iff__diff__eq__0,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = B2 )
    <=> ( aa_real_real(minus_minus_real(A2),B2) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_1394_eq__iff__diff__eq__0,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = B2 )
    <=> ( minus_minus_rat(A2,B2) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_1395_eq__iff__diff__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_1396_diff__eq__diff__less__eq,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A2),B2) = aa_real_real(minus_minus_real(C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1397_diff__eq__diff__less__eq,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A2,B2) = minus_minus_rat(C,D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1398_diff__eq__diff__less__eq,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1399_diff__right__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A2),C)),aa_real_real(minus_minus_real(B2),C)) ) ).

% diff_right_mono
tff(fact_1400_diff__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A2,C)),minus_minus_rat(B2,C)) ) ).

% diff_right_mono
tff(fact_1401_diff__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) ) ).

% diff_right_mono
tff(fact_1402_diff__left__mono,axiom,
    ! [B2: real,A2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(C),A2)),aa_real_real(minus_minus_real(C),B2)) ) ).

% diff_left_mono
tff(fact_1403_diff__left__mono,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(C,A2)),minus_minus_rat(C,B2)) ) ).

% diff_left_mono
tff(fact_1404_diff__left__mono,axiom,
    ! [B2: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2)) ) ).

% diff_left_mono
tff(fact_1405_diff__mono,axiom,
    ! [A2: real,B2: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A2),C)),aa_real_real(minus_minus_real(B2),D)) ) ) ).

% diff_mono
tff(fact_1406_diff__mono,axiom,
    ! [A2: rat,B2: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A2,C)),minus_minus_rat(B2,D)) ) ) ).

% diff_mono
tff(fact_1407_diff__mono,axiom,
    ! [A2: int,B2: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),D)) ) ) ).

% diff_mono
tff(fact_1408_diff__strict__right__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A2),C)),aa_real_real(minus_minus_real(B2),C)) ) ).

% diff_strict_right_mono
tff(fact_1409_diff__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A2,C)),minus_minus_rat(B2,C)) ) ).

% diff_strict_right_mono
tff(fact_1410_diff__strict__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) ) ).

% diff_strict_right_mono
tff(fact_1411_diff__strict__left__mono,axiom,
    ! [B2: real,A2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(C),A2)),aa_real_real(minus_minus_real(C),B2)) ) ).

% diff_strict_left_mono
tff(fact_1412_diff__strict__left__mono,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(C,A2)),minus_minus_rat(C,B2)) ) ).

% diff_strict_left_mono
tff(fact_1413_diff__strict__left__mono,axiom,
    ! [B2: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2)) ) ).

% diff_strict_left_mono
tff(fact_1414_diff__eq__diff__less,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A2),B2) = aa_real_real(minus_minus_real(C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_1415_diff__eq__diff__less,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( ( minus_minus_rat(A2,B2) = minus_minus_rat(C,D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_1416_diff__eq__diff__less,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_1417_diff__strict__mono,axiom,
    ! [A2: real,B2: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A2),C)),aa_real_real(minus_minus_real(B2),D)) ) ) ).

% diff_strict_mono
tff(fact_1418_diff__strict__mono,axiom,
    ! [A2: rat,B2: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A2,C)),minus_minus_rat(B2,D)) ) ) ).

% diff_strict_mono
tff(fact_1419_diff__strict__mono,axiom,
    ! [A2: int,B2: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),D)) ) ) ).

% diff_strict_mono
tff(fact_1420_exists__leI,axiom,
    ! [N: nat,P: fun_nat_o] :
      ( ( ! [N6: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N6),N)
           => ~ aa_nat_o(P,N6) )
       => aa_nat_o(P,N) )
     => ? [N7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N)
          & aa_nat_o(P,N7) ) ) ).

% exists_leI
tff(fact_1421_fstE,axiom,
    ! [Xb: product_prod_num_num,A2: num,B2: num,P: fun_num_o] :
      ( ( Xb = aa_num3156681643174609981um_num(product_Pair_num_num(A2),B2) )
     => ( aa_num_o(P,aa_Pro759708478137009011um_num(product_fst_num_num,Xb))
       => aa_num_o(P,A2) ) ) ).

% fstE
tff(fact_1422_fstE,axiom,
    ! [Xb: produc8398139464844984134T_VEBT,A2: nat,B2: produc4813437837504472865T_VEBT,P: fun_nat_o] :
      ( ( Xb = aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A2),B2) )
     => ( aa_nat_o(P,aa_Pro6878622105917549268BT_nat(produc758997459209783180T_VEBT,Xb))
       => aa_nat_o(P,A2) ) ) ).

% fstE
tff(fact_1423_fstE,axiom,
    ! [Xb: product_prod_nat_num,A2: nat,B2: num,P: fun_nat_o] :
      ( ( Xb = aa_num5026351239867140595at_num(product_Pair_nat_num(A2),B2) )
     => ( aa_nat_o(P,aa_Pro6261568864010571487um_nat(product_fst_nat_num,Xb))
       => aa_nat_o(P,A2) ) ) ).

% fstE
tff(fact_1424_fstE,axiom,
    ! [Xb: product_prod_nat_nat,A2: nat,B2: nat,P: fun_nat_o] :
      ( ( Xb = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2) )
     => ( aa_nat_o(P,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Xb))
       => aa_nat_o(P,A2) ) ) ).

% fstE
tff(fact_1425_fstE,axiom,
    ! [Xb: product_prod_int_int,A2: int,B2: int,P: fun_int_o] :
      ( ( Xb = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
     => ( aa_int_o(P,aa_Pro6799258434465214121nt_int(product_fst_int_int,Xb))
       => aa_int_o(P,A2) ) ) ).

% fstE
tff(fact_1426_fstE,axiom,
    ! [Xb: produc6575502325842934193n_assn,A2: assn,B2: assn,P: fun_assn_o] :
      ( ( Xb = aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(A2),B2) )
     => ( aa_assn_o(P,aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Xb))
       => aa_assn_o(P,A2) ) ) ).

% fstE
tff(fact_1427_sndE,axiom,
    ! [Xb: product_prod_num_num,A2: num,B2: num,P: fun_num_o] :
      ( ( Xb = aa_num3156681643174609981um_num(product_Pair_num_num(A2),B2) )
     => ( aa_num_o(P,aa_Pro759708478137009011um_num(product_snd_num_num,Xb))
       => aa_num_o(P,B2) ) ) ).

% sndE
tff(fact_1428_sndE,axiom,
    ! [Xb: produc8398139464844984134T_VEBT,A2: nat,B2: produc4813437837504472865T_VEBT,P: fun_Pr984851032239314682VEBT_o] :
      ( ( Xb = aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A2),B2) )
     => ( aa_Pro3461009536670225267VEBT_o(P,aa_Pro3664079603483438877T_VEBT(produc2084898568784432842T_VEBT,Xb))
       => aa_Pro3461009536670225267VEBT_o(P,B2) ) ) ).

% sndE
tff(fact_1429_sndE,axiom,
    ! [Xb: product_prod_nat_num,A2: nat,B2: num,P: fun_num_o] :
      ( ( Xb = aa_num5026351239867140595at_num(product_Pair_nat_num(A2),B2) )
     => ( aa_num_o(P,aa_Pro2818900650166350121um_num(product_snd_nat_num,Xb))
       => aa_num_o(P,B2) ) ) ).

% sndE
tff(fact_1430_sndE,axiom,
    ! [Xb: product_prod_nat_nat,A2: nat,B2: nat,P: fun_nat_o] :
      ( ( Xb = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2) )
     => ( aa_nat_o(P,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Xb))
       => aa_nat_o(P,B2) ) ) ).

% sndE
tff(fact_1431_sndE,axiom,
    ! [Xb: product_prod_int_int,A2: int,B2: int,P: fun_int_o] :
      ( ( Xb = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
     => ( aa_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,Xb))
       => aa_int_o(P,B2) ) ) ).

% sndE
tff(fact_1432_sndE,axiom,
    ! [Xb: produc6575502325842934193n_assn,A2: assn,B2: assn,P: fun_assn_o] :
      ( ( Xb = aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(A2),B2) )
     => ( aa_assn_o(P,aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Xb))
       => aa_assn_o(P,B2) ) ) ).

% sndE
tff(fact_1433_All__prod__contract,axiom,
    ! [P: fun_nat_fun_nat_o] :
      ( ! [A6: nat,X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A6),X_12)
    <=> ! [Z3: product_prod_nat_nat] : aa_nat_o(aa_nat_fun_nat_o(P,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Z3)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Z3)) ) ).

% All_prod_contract
tff(fact_1434_All__prod__contract,axiom,
    ! [P: fun_int_fun_int_o] :
      ( ! [A6: int,X_12: int] : aa_int_o(aa_int_fun_int_o(P,A6),X_12)
    <=> ! [Z3: product_prod_int_int] : aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_fst_int_int,Z3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Z3)) ) ).

% All_prod_contract
tff(fact_1435_All__prod__contract,axiom,
    ! [P: fun_assn_fun_assn_o] :
      ( ! [A6: assn,X_12: assn] : aa_assn_o(aa_assn_fun_assn_o(P,A6),X_12)
    <=> ! [Z3: produc6575502325842934193n_assn] : aa_assn_o(aa_assn_fun_assn_o(P,aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Z3)),aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Z3)) ) ).

% All_prod_contract
tff(fact_1436_Ex__prod__contract,axiom,
    ! [P: fun_nat_fun_nat_o] :
      ( ? [A6: nat,X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A6),X_12)
    <=> ? [Z3: product_prod_nat_nat] : aa_nat_o(aa_nat_fun_nat_o(P,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Z3)),aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Z3)) ) ).

% Ex_prod_contract
tff(fact_1437_Ex__prod__contract,axiom,
    ! [P: fun_int_fun_int_o] :
      ( ? [A6: int,X_12: int] : aa_int_o(aa_int_fun_int_o(P,A6),X_12)
    <=> ? [Z3: product_prod_int_int] : aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_fst_int_int,Z3)),aa_Pro6799258434465214121nt_int(product_snd_int_int,Z3)) ) ).

% Ex_prod_contract
tff(fact_1438_Ex__prod__contract,axiom,
    ! [P: fun_assn_fun_assn_o] :
      ( ? [A6: assn,X_12: assn] : aa_assn_o(aa_assn_fun_assn_o(P,A6),X_12)
    <=> ? [Z3: produc6575502325842934193n_assn] : aa_assn_o(aa_assn_fun_assn_o(P,aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,Z3)),aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,Z3)) ) ).

% Ex_prod_contract
tff(fact_1439_fn__fst__conv,axiom,
    ! [Fa: fun_na1476630412504255114at_nat] : aTP_Lamm_ho(Fa) = produc2626176000494625587at_nat(aTP_Lamm_hp(Fa)) ).

% fn_fst_conv
tff(fact_1440_fn__fst__conv,axiom,
    ! [Fa: fun_nat_o] : aTP_Lamm_hq(Fa) = produc6081775807080527818_nat_o(aTP_Lamm_hr(Fa)) ).

% fn_fst_conv
tff(fact_1441_fn__fst__conv,axiom,
    ! [Fa: fun_in4211310926729683942nt_int] : aTP_Lamm_hs(Fa) = produc4245557441103728435nt_int(aTP_Lamm_ht(Fa)) ).

% fn_fst_conv
tff(fact_1442_fn__fst__conv,axiom,
    ! [Fa: fun_int_o] : aTP_Lamm_hu(Fa) = produc4947309494688390418_int_o(aTP_Lamm_hv(Fa)) ).

% fn_fst_conv
tff(fact_1443_fn__fst__conv,axiom,
    ! [Fa: fun_int_int] : aTP_Lamm_hw(Fa) = produc8211389475949308722nt_int(aTP_Lamm_hx(Fa)) ).

% fn_fst_conv
tff(fact_1444_fn__snd__conv,axiom,
    ! [Fa: fun_na1476630412504255114at_nat] : aTP_Lamm_hy(Fa) = produc2626176000494625587at_nat(aTP_Lamm_hz(Fa)) ).

% fn_snd_conv
tff(fact_1445_fn__snd__conv,axiom,
    ! [Fa: fun_nat_o] : aTP_Lamm_ia(Fa) = produc6081775807080527818_nat_o(aTP_Lamm_ib(Fa)) ).

% fn_snd_conv
tff(fact_1446_fn__snd__conv,axiom,
    ! [Fa: fun_in4211310926729683942nt_int] : aTP_Lamm_ic(Fa) = produc4245557441103728435nt_int(aTP_Lamm_id(Fa)) ).

% fn_snd_conv
tff(fact_1447_fn__snd__conv,axiom,
    ! [Fa: fun_int_o] : aTP_Lamm_ie(Fa) = produc4947309494688390418_int_o(aTP_Lamm_if(Fa)) ).

% fn_snd_conv
tff(fact_1448_fn__snd__conv,axiom,
    ! [Fa: fun_int_int] : aTP_Lamm_ig(Fa) = produc8211389475949308722nt_int(aTP_Lamm_ih(Fa)) ).

% fn_snd_conv
tff(fact_1449_le__iff__diff__le__0,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A2),B2)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_1450_le__iff__diff__le__0,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A2,B2)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_1451_le__iff__diff__le__0,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_1452_less__iff__diff__less__0,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A2),B2)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_1453_less__iff__diff__less__0,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A2,B2)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_1454_less__iff__diff__less__0,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_1455_nat__compl__induct_H,axiom,
    ! [P: fun_nat_o,N: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N3: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N3)
               => aa_nat_o(P,Nn) )
           => aa_nat_o(P,aa_nat_nat(suc,N3)) )
       => aa_nat_o(P,N) ) ) ).

% nat_compl_induct'
tff(fact_1456_nat__compl__induct,axiom,
    ! [P: fun_nat_o,N: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N3: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N3)
               => aa_nat_o(P,Nn) )
           => aa_nat_o(P,aa_nat_nat(suc,N3)) )
       => aa_nat_o(P,N) ) ) ).

% nat_compl_induct
tff(fact_1457_nat__in__between__eq_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(suc,A2)) )
    <=> ( B2 = aa_nat_nat(suc,A2) ) ) ).

% nat_in_between_eq(1)
tff(fact_1458_nat__in__between__eq_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),aa_nat_nat(suc,A2)) )
    <=> ( B2 = A2 ) ) ).

% nat_in_between_eq(2)
tff(fact_1459_Suc__to__right,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(suc,N) = M )
     => ( N = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat)) ) ) ).

% Suc_to_right
tff(fact_1460_nat__geq__1__eq__neqz,axiom,
    ! [Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Xb)
    <=> ( Xb != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
tff(fact_1461_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_VE6480751511521678896_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),N)
         => ? [Li: vEBT_VEBT] : aa_nat_o(aa_VEB2314620471664397097_nat_o(P,Li),I2) )
     => ~ ! [L2: list_VEBT_VEBT] :
            ( ( size_s6755466524823107622T_VEBT(L2) = N )
           => ~ ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N)
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(P,nth_VEBT_VEBT(L2,I3)),I3) ) ) ) ).

% obtain_list_from_elements
tff(fact_1462_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_VE6346650636051888901_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),N)
         => ? [Li: vEBT_VEBTi] : aa_nat_o(aa_VEB8883426185042993134_nat_o(P,Li),I2) )
     => ~ ! [L2: list_VEBT_VEBTi] :
            ( ( size_s7982070591426661849_VEBTi(L2) = N )
           => ~ ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N)
                 => aa_nat_o(aa_VEB8883426185042993134_nat_o(P,nth_VEBT_VEBTi(L2,I3)),I3) ) ) ) ).

% obtain_list_from_elements
tff(fact_1463_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_real_fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),N)
         => ? [Li: real] : aa_nat_o(aa_real_fun_nat_o(P,Li),I2) )
     => ~ ! [L2: list_real] :
            ( ( size_size_list_real(L2) = N )
           => ~ ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N)
                 => aa_nat_o(aa_real_fun_nat_o(P,aa_nat_real(nth_real(L2),I3)),I3) ) ) ) ).

% obtain_list_from_elements
tff(fact_1464_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_o_fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),N)
         => ? [Li: $o] : aa_nat_o(aa_o_fun_nat_o(P,(Li)),I2) )
     => ~ ! [L2: list_o] :
            ( ( size_size_list_o(L2) = N )
           => ~ ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N)
                 => aa_nat_o(aa_o_fun_nat_o(P,aa_nat_o(nth_o(L2),I3)),I3) ) ) ) ).

% obtain_list_from_elements
tff(fact_1465_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_nat_fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),N)
         => ? [Li: nat] : aa_nat_o(aa_nat_fun_nat_o(P,Li),I2) )
     => ~ ! [L2: list_nat] :
            ( ( size_size_list_nat(L2) = N )
           => ~ ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N)
                 => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(nth_nat(L2),I3)),I3) ) ) ) ).

% obtain_list_from_elements
tff(fact_1466_obtain__list__from__elements,axiom,
    ! [N: nat,P: fun_int_fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),N)
         => ? [Li: int] : aa_nat_o(aa_int_fun_nat_o(P,Li),I2) )
     => ~ ! [L2: list_int] :
            ( ( size_size_list_int(L2) = N )
           => ~ ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),N)
                 => aa_nat_o(aa_int_fun_nat_o(P,aa_nat_int(nth_int(L2),I3)),I3) ) ) ) ).

% obtain_list_from_elements
tff(fact_1467_vebt__insert_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S) ).

% vebt_insert.simps(2)
tff(fact_1468_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv2),Uw2)),Xb) ).

% vebt_member.simps(2)
tff(fact_1469_le__some__optE,axiom,
    ! [M: set_nat,Xb: option_set_nat] :
      ( ord_le2843612097646854710et_nat(some_set_nat(M),Xb)
     => ~ ! [M7: set_nat] :
            ( ( Xb = some_set_nat(M7) )
           => ~ aa_set_nat_o(ord_less_eq_set_nat(M),M7) ) ) ).

% le_some_optE
tff(fact_1470_le__some__optE,axiom,
    ! [M: rat,Xb: option_rat] :
      ( ord_le2406147912482264968on_rat(some_rat(M),Xb)
     => ~ ! [M7: rat] :
            ( ( Xb = some_rat(M7) )
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),M7) ) ) ).

% le_some_optE
tff(fact_1471_le__some__optE,axiom,
    ! [M: num,Xb: option_num] :
      ( aa_option_num_o(aa_opt8674010985164783839_num_o(ord_le6622620407824499402on_num,aa_num_option_num(some_num,M)),Xb)
     => ~ ! [M7: num] :
            ( ( Xb = aa_num_option_num(some_num,M7) )
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),M7) ) ) ).

% le_some_optE
tff(fact_1472_le__some__optE,axiom,
    ! [M: nat,Xb: option_nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,aa_nat_option_nat(some_nat,M)),Xb)
     => ~ ! [M7: nat] :
            ( ( Xb = aa_nat_option_nat(some_nat,M7) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),M7) ) ) ).

% le_some_optE
tff(fact_1473_le__some__optE,axiom,
    ! [M: int,Xb: option_int] :
      ( ord_le1736525451366464988on_int(aa_int_option_int(some_int,M),Xb)
     => ~ ! [M7: int] :
            ( ( Xb = aa_int_option_int(some_int,M7) )
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),M7) ) ) ).

% le_some_optE
tff(fact_1474_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
     => ( ! [A: $o,B: $o] :
            ( ( Xb = vEBT_Leaf((A),(B)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A),
                  $ite(Xaa = one_one_nat,(B),$false) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)
             => ~ $ite(
                    Xaa = Mi2,
                    $true,
                    $ite(
                      Xaa = Ma2,
                      $true,
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                        $false,
                        $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_1475_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
      <=> (Y) )
     => ( ! [A: $o,B: $o] :
            ( ( Xb = vEBT_Leaf((A),(B)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A),
                    $ite(Xaa = one_one_nat,(B),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)
           => (Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)
             => (Y) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)
               => (Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi2,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_1476_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
     => ( ! [A: $o,B: $o] :
            ( ( Xb = vEBT_Leaf((A),(B)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A),
                $ite(Xaa = one_one_nat,(B),$false) ) )
       => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)
                   => $ite(
                        Xaa = Mi2,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_1477_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o] :
      ( ! [X: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X,set_VEBT_VEBTi2(L))
         => aa_VEBT_VEBTi_o(P,X) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s7982070591426661849_VEBTi(L))
         => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(L,I4)) ) ) ).

% all_set_conv_nth
tff(fact_1478_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(L))
         => aa_VEBT_VEBT_o(P,X) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6755466524823107622T_VEBT(L))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(L,I4)) ) ) ).

% all_set_conv_nth
tff(fact_1479_all__set__conv__nth,axiom,
    ! [L: list_P8527749157015355191n_assn,P: fun_Pr3571187171380813162assn_o] :
      ( ! [X: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(L))
         => aa_Pro2491721552408928739assn_o(P,X) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6829681357464350627n_assn(L))
         => aa_Pro2491721552408928739assn_o(P,nth_Pr1769885009046257848n_assn(L,I4)) ) ) ).

% all_set_conv_nth
tff(fact_1480_all__set__conv__nth,axiom,
    ! [L: list_real,P: fun_real_o] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(L))
         => aa_real_o(P,X) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_real(L))
         => aa_real_o(P,aa_nat_real(nth_real(L),I4)) ) ) ).

% all_set_conv_nth
tff(fact_1481_all__set__conv__nth,axiom,
    ! [L: list_o,P: fun_o_o] :
      ( ! [X: $o] :
          ( member_o((X),set_o2(L))
         => aa_o_o(P,(X)) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_o(L))
         => aa_o_o(P,aa_nat_o(nth_o(L),I4)) ) ) ).

% all_set_conv_nth
tff(fact_1482_all__set__conv__nth,axiom,
    ! [L: list_nat,P: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(L))
         => aa_nat_o(P,X) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_nat(L))
         => aa_nat_o(P,aa_nat_nat(nth_nat(L),I4)) ) ) ).

% all_set_conv_nth
tff(fact_1483_all__set__conv__nth,axiom,
    ! [L: list_int,P: fun_int_o] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(L))
         => aa_int_o(P,X) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_int(L))
         => aa_int_o(P,aa_nat_int(nth_int(L),I4)) ) ) ).

% all_set_conv_nth
tff(fact_1484_vebt__insert_Osimps_I3_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S) ).

% vebt_insert.simps(3)
tff(fact_1485_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz)),Xb) ).

% vebt_member.simps(3)
tff(fact_1486_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),aa_nat_nat(suc,zero_zero_nat))),M) ) ) ).

% nz_le_conv_less
tff(fact_1487_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_1488_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)),Xb) ).

% vebt_member.simps(4)
tff(fact_1489_vebt__maxt_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( ! [A: $o,B: $o] :
            ( ( Xb = vEBT_Leaf((A),(B)) )
           => ( Y != $ite(
                  (B),
                  aa_nat_option_nat(some_nat,one_one_nat),
                  $ite((A),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2)
               => ( Y != aa_nat_option_nat(some_nat,Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_1490_vebt__mint_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( ! [A: $o,B: $o] :
            ( ( Xb = vEBT_Leaf((A),(B)) )
           => ( Y != $ite(
                  (A),
                  aa_nat_option_nat(some_nat,zero_zero_nat),
                  $ite((B),aa_nat_option_nat(some_nat,one_one_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2)
               => ( Y != aa_nat_option_nat(some_nat,Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_1491_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% zle_diff1_eq
tff(fact_1492_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] :
      ( ( aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Y11,Y12,Y13),Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_1493_minus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,K),zero_zero_int) = K ).

% minus_int_code(1)
tff(fact_1494_VEBT_Ocase__distrib,axiom,
    ! [H: fun_Pr3650993782763339253T_VEBT,F1: fun_op8229906066235313459T_VEBT,F2: fun_o_8142510167905321106T_VEBT,VEBT: vEBT_VEBT] : aa_Pro5421721056952420150T_VEBT(H,vEBT_c634343235235684882T_VEBT(F1,F2,VEBT)) = vEBT_c634343235235684882T_VEBT(aa_fun7544404396881244796T_VEBT(aTP_Lamm_ii(H),F1),aa_fun5380946117177639010T_VEBT(aTP_Lamm_ij(H),F2),VEBT) ).

% VEBT.case_distrib
tff(fact_1495_VEBT_Ocase__distrib,axiom,
    ! [H: fun_Pr7471953115884902709VEBT_o,F1: fun_op8229906066235313459T_VEBT,F2: fun_o_8142510167905321106T_VEBT,VEBT: vEBT_VEBT] :
      ( aa_Pro5475924895142249886VEBT_o(H,vEBT_c634343235235684882T_VEBT(F1,F2,VEBT))
    <=> aa_VEBT_VEBT_o(vEBT_case_VEBT_o(aa_fun5124765266653172784VEBT_o(aTP_Lamm_ik(H),F1),aa_fun449888776569364894un_o_o(aTP_Lamm_il(H),F2)),VEBT) ) ).

% VEBT.case_distrib
tff(fact_1496_VEBT_Ocase__distrib,axiom,
    ! [H: fun_o_3001263932818058983T_VEBT,F1: fun_op541075054904862521VEBT_o,F2: fun_o_fun_o_o,VEBT: vEBT_VEBT] : aa_o_P8223745691486049304T_VEBT(H,aa_VEBT_VEBT_o(vEBT_case_VEBT_o(F1,F2),VEBT)) = vEBT_c634343235235684882T_VEBT(aa_fun8362160726713724924T_VEBT(aTP_Lamm_im(H),F1),aa_fun4185041498930275652T_VEBT(aTP_Lamm_in(H),F2),VEBT) ).

% VEBT.case_distrib
tff(fact_1497_VEBT_Ocase__distrib,axiom,
    ! [H: fun_o_o,F1: fun_op541075054904862521VEBT_o,F2: fun_o_fun_o_o,VEBT: vEBT_VEBT] :
      ( aa_o_o(H,aa_VEBT_VEBT_o(vEBT_case_VEBT_o(F1,F2),VEBT))
    <=> aa_VEBT_VEBT_o(vEBT_case_VEBT_o(aa_fun2455880738235528880VEBT_o(aTP_Lamm_io(H),F1),aa_fun6846671739589424444un_o_o(aTP_Lamm_ip(H),F2)),VEBT) ) ).

% VEBT.case_distrib
tff(fact_1498_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc650719256901721312it_nat] :
      ( ! [Uu2: fun_Pr226060869445836462_nat_o,Uv: option2621746655072343315it_nat] : Xb != produc4438071236943126352it_nat(Uu2,produc6851560022941992023it_nat(none_P1551326421579882414it_nat,Uv))
     => ( ! [Uw: fun_Pr226060869445836462_nat_o,V2: produc120671012495760973it_nat] : Xb != produc4438071236943126352it_nat(Uw,produc6851560022941992023it_nat(some_P2407035485129114418it_nat(V2),none_P1551326421579882414it_nat))
       => ~ ! [F3: fun_Pr226060869445836462_nat_o,X2: produc120671012495760973it_nat,Y2: produc120671012495760973it_nat] : Xb != produc4438071236943126352it_nat(F3,produc6851560022941992023it_nat(some_P2407035485129114418it_nat(X2),some_P2407035485129114418it_nat(Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1499_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc3124918679657246250it_nat] :
      ( ! [Uu2: fun_Pr5882989906983993892_nat_o,Uv: option7339022715339332451it_nat] : Xb != produc5062306165551350682it_nat(Uu2,produc9206348758962449759it_nat(none_P7668321371905463026it_nat,Uv))
     => ( ! [Uw: fun_Pr5882989906983993892_nat_o,V2: produc8047831477865546771it_nat] : Xb != produc5062306165551350682it_nat(Uw,produc9206348758962449759it_nat(some_P468703482102919278it_nat(V2),none_P7668321371905463026it_nat))
       => ~ ! [F3: fun_Pr5882989906983993892_nat_o,X2: produc8047831477865546771it_nat,Y2: produc8047831477865546771it_nat] : Xb != produc5062306165551350682it_nat(F3,produc9206348758962449759it_nat(some_P468703482102919278it_nat(X2),some_P468703482102919278it_nat(Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1500_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc1615844007061411358on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_o,Uv: option_nat] : Xb != produc7967433646104525454on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv))
     => ( ! [Uw: fun_nat_fun_nat_o,V2: nat] : Xb != produc7967433646104525454on_nat(Uw,produc5098337634421038937on_nat(aa_nat_option_nat(some_nat,V2),none_nat))
       => ~ ! [F3: fun_nat_fun_nat_o,X2: nat,Y2: nat] : Xb != produc7967433646104525454on_nat(F3,produc5098337634421038937on_nat(aa_nat_option_nat(some_nat,X2),aa_nat_option_nat(some_nat,Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1501_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc3739950874144486240at_nat] :
      ( ! [Uu2: fun_Pr2028404973800837934_nat_o,Uv: option4927543243414619207at_nat] : Xb != produc3653971535690468304at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv))
     => ( ! [Uw: fun_Pr2028404973800837934_nat_o,V2: product_prod_nat_nat] : Xb != produc3653971535690468304at_nat(Uw,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F3: fun_Pr2028404973800837934_nat_o,X2: product_prod_nat_nat,Y2: product_prod_nat_nat] : Xb != produc3653971535690468304at_nat(F3,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X2),some_P7363390416028606310at_nat(Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1502_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc50614890452615238on_num] :
      ( ! [Uu2: fun_num_fun_num_o,Uv: option_num] : Xb != produc1140783415258738358on_num(Uu2,produc8585076106096196333on_num(none_num,Uv))
     => ( ! [Uw: fun_num_fun_num_o,V2: num] : Xb != produc1140783415258738358on_num(Uw,produc8585076106096196333on_num(aa_num_option_num(some_num,V2),none_num))
       => ~ ! [F3: fun_num_fun_num_o,X2: num,Y2: num] : Xb != produc1140783415258738358on_num(F3,produc8585076106096196333on_num(aa_num_option_num(some_num,X2),aa_num_option_num(some_num,Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1503_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc1531648545593279979it_nat] :
      ( ! [Uu2: fun_Pr3118856740240616547it_nat,Uv: option2621746655072343315it_nat] : Xb != produc2260137262690574555it_nat(Uu2,produc6851560022941992023it_nat(none_P1551326421579882414it_nat,Uv))
     => ( ! [Uw: fun_Pr3118856740240616547it_nat,V2: produc120671012495760973it_nat] : Xb != produc2260137262690574555it_nat(Uw,produc6851560022941992023it_nat(some_P2407035485129114418it_nat(V2),none_P1551326421579882414it_nat))
       => ~ ! [F3: fun_Pr3118856740240616547it_nat,A: produc120671012495760973it_nat,B: produc120671012495760973it_nat] : Xb != produc2260137262690574555it_nat(F3,produc6851560022941992023it_nat(some_P2407035485129114418it_nat(A),some_P2407035485129114418it_nat(B))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1504_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc7847833613904905897it_nat] :
      ( ! [Uu2: fun_Pr1313076919630728883it_nat,Uv: option7339022715339332451it_nat] : Xb != produc5529849720642941531it_nat(Uu2,produc9206348758962449759it_nat(none_P7668321371905463026it_nat,Uv))
     => ( ! [Uw: fun_Pr1313076919630728883it_nat,V2: produc8047831477865546771it_nat] : Xb != produc5529849720642941531it_nat(Uw,produc9206348758962449759it_nat(some_P468703482102919278it_nat(V2),none_P7668321371905463026it_nat))
       => ~ ! [F3: fun_Pr1313076919630728883it_nat,A: produc8047831477865546771it_nat,B: produc8047831477865546771it_nat] : Xb != produc5529849720642941531it_nat(F3,produc9206348758962449759it_nat(some_P468703482102919278it_nat(A),some_P468703482102919278it_nat(B))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1505_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc7004362771917097900on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_nat,Uv: option_nat] : Xb != produc6699827698519632094on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv))
     => ( ! [Uw: fun_nat_fun_nat_nat,V2: nat] : Xb != produc6699827698519632094on_nat(Uw,produc5098337634421038937on_nat(aa_nat_option_nat(some_nat,V2),none_nat))
       => ~ ! [F3: fun_nat_fun_nat_nat,A: nat,B: nat] : Xb != produc6699827698519632094on_nat(F3,produc5098337634421038937on_nat(aa_nat_option_nat(some_nat,A),aa_nat_option_nat(some_nat,B))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1506_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc1260151622200656695at_nat] :
      ( ! [Uu2: fun_Pr2477239776880934807at_nat,Uv: option4927543243414619207at_nat] : Xb != produc7814092408141220007at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv))
     => ( ! [Uw: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : Xb != produc7814092408141220007at_nat(Uw,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F3: fun_Pr2477239776880934807at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] : Xb != produc7814092408141220007at_nat(F3,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A),some_P7363390416028606310at_nat(B))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1507_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc2703417854250991842on_num] :
      ( ! [Uu2: fun_num_fun_num_num,Uv: option_num] : Xb != produc6360833703471310868on_num(Uu2,produc8585076106096196333on_num(none_num,Uv))
     => ( ! [Uw: fun_num_fun_num_num,V2: num] : Xb != produc6360833703471310868on_num(Uw,produc8585076106096196333on_num(aa_num_option_num(some_num,V2),none_num))
       => ~ ! [F3: fun_num_fun_num_num,A: num,B: num] : Xb != produc6360833703471310868on_num(F3,produc8585076106096196333on_num(aa_num_option_num(some_num,A),aa_num_option_num(some_num,B))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1508_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_1509_less__eq__int__code_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% less_eq_int_code(1)
tff(fact_1510_less__int__code_I1_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_int_code(1)
tff(fact_1511_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K)
     => ( aa_int_o(P,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_le_induct
tff(fact_1512_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_less_induct
tff(fact_1513_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv: $o,D2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv))),D2)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList2),Summary)),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_1514_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(X112,X122,X132),X142)
     => ~ ! [X212: $o,X223: $o] : Y != vEBT_Leaf((X212),(X223)) ) ).

% VEBT.exhaust
tff(fact_1515_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] : aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14) != vEBT_Leaf((X21),(X222)) ).

% VEBT.distinct(1)
tff(fact_1516_VEBT_Odisc_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_is_Node,aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14)) ).

% VEBT.disc(1)
tff(fact_1517_VEBT_OdiscI_I1_J,axiom,
    ! [VEBT: vEBT_VEBT,X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( VEBT = aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14) )
     => aa_VEBT_VEBT_o(vEBT_is_Node,VEBT) ) ).

% VEBT.discI(1)
tff(fact_1518_is__Node__def,axiom,
    ! [VEBT: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_is_Node,VEBT)
    <=> ? [X113: option4927543243414619207at_nat,X123: nat,X133: list_VEBT_VEBT,X143: vEBT_VEBT] : VEBT = aa_VEB5832953974749077948T_VEBT(vEBT_Node(X113,X123,X133),X143) ) ).

% is_Node_def
tff(fact_1519_VEBT_Osimps_I5_J,axiom,
    ! [F1: fun_op8229906066235313459T_VEBT,F2: fun_o_8142510167905321106T_VEBT,X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : vEBT_c634343235235684882T_VEBT(F1,F2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14)) = aa_VEB8692087746923902260T_VEBT(aa_lis2966823273985285125T_VEBT(aa_nat6705503460730637082T_VEBT(aa_opt2902932101092860132T_VEBT(F1,X11),X12),X13),X14) ).

% VEBT.simps(5)
tff(fact_1520_VEBT_Osimps_I5_J,axiom,
    ! [F1: fun_op541075054904862521VEBT_o,F2: fun_o_fun_o_o,X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_case_VEBT_o(F1,F2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14))
    <=> aa_VEBT_VEBT_o(aa_lis4184305612057247525VEBT_o(aa_nat8798449810042143974VEBT_o(aa_opt7072554309604550194VEBT_o(F1,X11),X12),X13),X14) ) ).

% VEBT.simps(5)
tff(fact_1521_VEBT_Odisc_I2_J,axiom,
    ! [X21: $o,X222: $o] : ~ aa_VEBT_VEBT_o(vEBT_is_Node,vEBT_Leaf((X21),(X222))) ).

% VEBT.disc(2)
tff(fact_1522_VEBT_Osimps_I6_J,axiom,
    ! [F1: fun_op8229906066235313459T_VEBT,F2: fun_o_8142510167905321106T_VEBT,X21: $o,X222: $o] : vEBT_c634343235235684882T_VEBT(F1,F2,vEBT_Leaf((X21),(X222))) = aa_o_P8223745691486049304T_VEBT(aa_o_f7062522455407347339T_VEBT(F2,(X21)),(X222)) ).

% VEBT.simps(6)
tff(fact_1523_VEBT_Osimps_I6_J,axiom,
    ! [F1: fun_op541075054904862521VEBT_o,F2: fun_o_fun_o_o,X21: $o,X222: $o] :
      ( aa_VEBT_VEBT_o(vEBT_case_VEBT_o(F1,F2),vEBT_Leaf((X21),(X222)))
    <=> aa_o_o(aa_o_fun_o_o(F2,(X21)),(X222)) ) ).

% VEBT.simps(6)
tff(fact_1524_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [Fa: fun_Pr2477239776880934807at_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(Fa,some_P7363390416028606310at_nat(A2),some_P7363390416028606310at_nat(B2)) = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(Fa,A2),B2)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1525_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [Fa: fun_num_fun_num_num,A2: num,B2: num] : aa_opt5369159329878751634on_num(vEBT_V819420779217536731ft_num(Fa,aa_num_option_num(some_num,A2)),aa_num_option_num(some_num,B2)) = aa_num_option_num(some_num,aa_num_num(aa_num_fun_num_num(Fa,A2),B2)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1526_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [Fa: fun_nat_fun_nat_nat,A2: nat,B2: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Fa),aa_nat_option_nat(some_nat,A2)),aa_nat_option_nat(some_nat,B2)) = aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(Fa,A2),B2)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1527_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_Pr3118856740240616547it_nat,Uv2: option2621746655072343315it_nat] : vEBT_V819568868292977612it_nat(Uu,none_P1551326421579882414it_nat,Uv2) = none_P1551326421579882414it_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_1528_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_Pr1313076919630728883it_nat,Uv2: option7339022715339332451it_nat] : vEBT_V613753007643960916it_nat(Uu,none_P7668321371905463026it_nat,Uv2) = none_P7668321371905463026it_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_1529_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : vEBT_V1502963449132264192at_nat(Uu,none_P5556105721700978146at_nat,Uv2) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_1530_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_num_fun_num_num,Uv2: option_num] : aa_opt5369159329878751634on_num(vEBT_V819420779217536731ft_num(Uu,none_num),Uv2) = none_num ).

% VEBT_internal.option_shift.simps(1)
tff(fact_1531_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uv2: option_nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uu),none_nat),Uv2) = none_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_1532_set__vebt__def,axiom,
    ! [Tb: vEBT_VEBT] : vEBT_set_vebt(Tb) = collect_nat(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Tb)) ).

% set_vebt_def
tff(fact_1533_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_Pr3118856740240616547it_nat,V: produc120671012495760973it_nat] : vEBT_V819568868292977612it_nat(Uw2,some_P2407035485129114418it_nat(V),none_P1551326421579882414it_nat) = none_P1551326421579882414it_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1534_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_Pr1313076919630728883it_nat,V: produc8047831477865546771it_nat] : vEBT_V613753007643960916it_nat(Uw2,some_P468703482102919278it_nat(V),none_P7668321371905463026it_nat) = none_P7668321371905463026it_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1535_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_Pr2477239776880934807at_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)
tff(fact_1536_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_num_fun_num_num,V: num] : aa_opt5369159329878751634on_num(vEBT_V819420779217536731ft_num(Uw2,aa_num_option_num(some_num,V)),none_num) = none_num ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1537_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: fun_nat_fun_nat_nat,V: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uw2),aa_nat_option_nat(some_nat,V)),none_nat) = none_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1538_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_Pr3118856740240616547it_nat,Xaa: option2621746655072343315it_nat,Xba: option2621746655072343315it_nat,Y: option2621746655072343315it_nat] :
      ( ( vEBT_V819568868292977612it_nat(Xb,Xaa,Xba) = Y )
     => ( ( ( Xaa = none_P1551326421579882414it_nat )
         => ( Y != none_P1551326421579882414it_nat ) )
       => ( ( ? [V2: produc120671012495760973it_nat] : Xaa = some_P2407035485129114418it_nat(V2)
           => ( ( Xba = none_P1551326421579882414it_nat )
             => ( Y != none_P1551326421579882414it_nat ) ) )
         => ~ ! [A: produc120671012495760973it_nat] :
                ( ( Xaa = some_P2407035485129114418it_nat(A) )
               => ! [B: produc120671012495760973it_nat] :
                    ( ( Xba = some_P2407035485129114418it_nat(B) )
                   => ( Y != some_P2407035485129114418it_nat(aa_Pro6543753689527670780it_nat(aa_Pro329082998274276628it_nat(Xb,A),B)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1539_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_Pr1313076919630728883it_nat,Xaa: option7339022715339332451it_nat,Xba: option7339022715339332451it_nat,Y: option7339022715339332451it_nat] :
      ( ( vEBT_V613753007643960916it_nat(Xb,Xaa,Xba) = Y )
     => ( ( ( Xaa = none_P7668321371905463026it_nat )
         => ( Y != none_P7668321371905463026it_nat ) )
       => ( ( ? [V2: produc8047831477865546771it_nat] : Xaa = some_P468703482102919278it_nat(V2)
           => ( ( Xba = none_P7668321371905463026it_nat )
             => ( Y != none_P7668321371905463026it_nat ) ) )
         => ~ ! [A: produc8047831477865546771it_nat] :
                ( ( Xaa = some_P468703482102919278it_nat(A) )
               => ! [B: produc8047831477865546771it_nat] :
                    ( ( Xba = some_P468703482102919278it_nat(B) )
                   => ( Y != some_P468703482102919278it_nat(aa_Pro7661135075737848164it_nat(aa_Pro52009495076012404it_nat(Xb,A),B)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1540_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_Pr2477239776880934807at_nat,Xaa: option4927543243414619207at_nat,Xba: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(Xb,Xaa,Xba) = Y )
     => ( ( ( Xaa = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] : Xaa = some_P7363390416028606310at_nat(V2)
           => ( ( Xba = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A: product_prod_nat_nat] :
                ( ( Xaa = some_P7363390416028606310at_nat(A) )
               => ! [B: product_prod_nat_nat] :
                    ( ( Xba = some_P7363390416028606310at_nat(B) )
                   => ( Y != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(Xb,A),B)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1541_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_num_fun_num_num,Xaa: option_num,Xba: option_num,Y: option_num] :
      ( ( aa_opt5369159329878751634on_num(vEBT_V819420779217536731ft_num(Xb,Xaa),Xba) = Y )
     => ( ( ( Xaa = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V2: num] : Xaa = aa_num_option_num(some_num,V2)
           => ( ( Xba = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A: num] :
                ( ( Xaa = aa_num_option_num(some_num,A) )
               => ! [B: num] :
                    ( ( Xba = aa_num_option_num(some_num,B) )
                   => ( Y != aa_num_option_num(some_num,aa_num_num(aa_num_fun_num_num(Xb,A),B)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1542_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: option_nat,Xba: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Xb),Xaa),Xba) = Y )
     => ( ( ( Xaa = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] : Xaa = aa_nat_option_nat(some_nat,V2)
           => ( ( Xba = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A: nat] :
                ( ( Xaa = aa_nat_option_nat(some_nat,A) )
               => ! [B: nat] :
                    ( ( Xba = aa_nat_option_nat(some_nat,B) )
                   => ( Y != aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(Xb,A),B)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1543_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A: $o,B: $o,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),X2)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv),Uw)),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2)),X2) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_1544_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_1545_invar__vebt_Ointros_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_Leaf((A2),(B2))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_1546_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,
    ! [Xb: vEBT_VEBT] :
      ( ( Xb != vEBT_Leaf($false,$false) )
     => ( ! [Uv: $o] : Xb != vEBT_Leaf($true,(Uv))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2),Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_1547_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : vEBT_vebt_mint(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv2),Uw2)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_1548_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : vEBT_vebt_maxt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv2),Uw2)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_1549_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A: $o,B: $o] : Xb != vEBT_Leaf((A),(B))
     => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_1550_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_mint(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux,Uy),Uz)) = aa_nat_option_nat(some_nat,Mi) ).

% vebt_mint.simps(3)
tff(fact_1551_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_maxt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux,Uy),Uz)) = aa_nat_option_nat(some_nat,Ma) ).

% vebt_maxt.simps(3)
tff(fact_1552_vebt__mint_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_mint(vEBT_Leaf((A2),(B2))) = $ite(
        (A2),
        aa_nat_option_nat(some_nat,zero_zero_nat),
        $ite((B2),aa_nat_option_nat(some_nat,one_one_nat),none_nat) ) ).

% vebt_mint.simps(1)
tff(fact_1553_vebt__maxt_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_maxt(vEBT_Leaf((A2),(B2))) = $ite(
        (B2),
        aa_nat_option_nat(some_nat,one_one_nat),
        $ite((A2),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) ) ).

% vebt_maxt.simps(1)
tff(fact_1554_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv: $o,Uw: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv))),Uw)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,Va3),Vb2)),X2)
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)),X2)
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)),X2) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_1555_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv: $o] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv))),zero_zero_nat)
     => ( ! [A: $o,Uw: $o] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(Uw))),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A: $o,B: $o,Va2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)),Vb2)
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)),Vj2)
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),X2) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
tff(fact_1556_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B: $o] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B))),zero_zero_nat)
     => ( ! [Uv: $o,Uw: $o,N3: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw))),aa_nat_nat(suc,N3))
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)),Va3)
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(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] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)),Vi2)
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),X2) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_1557_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,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A: $o,B: $o,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),X2)
     => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2)),X2)
       => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2)),X2)
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary)),X2)
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),X2) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_1558_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,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A: $o,B: $o,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),X2)
     => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)),X2)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)),X2)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)),X2)
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] : Xb != aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),X2) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_1559_atLeastatMost__subset__iff,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(A2,B2),set_or4548717258645045905et_nat(C,D))
    <=> ( ~ aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
        | ( aa_set_nat_o(ord_less_eq_set_nat(C),A2)
          & aa_set_nat_o(ord_less_eq_set_nat(B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_1560_atLeastatMost__subset__iff,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or633870826150836451st_rat(A2,B2)),set_or633870826150836451st_rat(C,D))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_1561_atLeastatMost__subset__iff,axiom,
    ! [A2: num,B2: num,C: num,D: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or7049704709247886629st_num(A2,B2)),set_or7049704709247886629st_num(C,D))
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A2)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_1562_atLeastatMost__subset__iff,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A2),B2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,C),D))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_1563_atLeastatMost__subset__iff,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or1266510415728281911st_int(A2,B2)),set_or1266510415728281911st_int(C,D))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_1564_atLeastatMost__subset__iff,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or1222579329274155063t_real(A2,B2)),set_or1222579329274155063t_real(C,D))
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_1565_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,Xb: nat] : vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm) ).

% vebt_delete.simps(6)
tff(fact_1566_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,A1),A22)
     => ( ( ? [A: $o,B: $o] : A1 = vEBT_Leaf((A),(B))
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary: vEBT_VEBT,M3: nat,Deg: nat] :
              ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2),Summary) )
             => ( ( A22 = Deg )
               => ( ! [X3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList2))
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),N3) )
                 => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M3)
                   => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3) )
                     => ( ( M3 = N3 )
                       => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M3) )
                         => ( ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X_13)
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList2))
                                 => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_13) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary: vEBT_VEBT,M3: nat,Deg: nat] :
                ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2),Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X3: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList2))
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),N3) )
                   => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M3)
                     => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3) )
                       => ( ( M3 = aa_nat_nat(suc,N3) )
                         => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M3) )
                           => ( ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X_13)
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_13) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary: vEBT_VEBT,M3: nat,Deg: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Deg,TreeList2),Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X3: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList2))
                         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),N3) )
                     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M3)
                       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3) )
                         => ( ( M3 = N3 )
                           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M3) )
                             => ( ! [I3: nat] :
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3))
                                   => ( ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),X_12)
                                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),I3) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X3: vEBT_VEBT] :
                                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList2))
                                       => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_13) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I3: nat] :
                                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I3 )
                                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N3)) )
                                                & ! [X3: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N3) = I3 )
                                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N3)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X3)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N3: nat,Summary: vEBT_VEBT,M3: nat,Deg: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Deg,TreeList2),Summary) )
                   => ( ( A22 = Deg )
                     => ( ! [X3: vEBT_VEBT] :
                            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList2))
                           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),N3) )
                       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M3)
                         => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3) )
                           => ( ( M3 = aa_nat_nat(suc,N3) )
                             => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),M3) )
                               => ( ! [I3: nat] :
                                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3))
                                     => ( ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),X_12)
                                      <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),I3) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X3: vEBT_VEBT] :
                                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeList2))
                                         => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_13) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I3: nat] :
                                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N3) = I3 )
                                                   => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N3)) )
                                                  & ! [X3: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N3) = I3 )
                                                        & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList2,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N3)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X3)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_1567_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,A1),A22)
    <=> ( ( ? [A6: $o,B6: $o] : A1 = vEBT_Leaf((A6),(B6))
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList3: list_VEBT_VEBT,N8: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3),Summary3) )
            & ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),N8) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary3),N8)
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N8),N8) )
            & ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary3),X_12)
            & ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_12) ) )
        | ? [TreeList3: list_VEBT_VEBT,N8: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3),Summary3) )
            & ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),N8) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary3),aa_nat_nat(suc,N8))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N8)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N8),aa_nat_nat(suc,N8)) )
            & ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary3),X_12)
            & ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_12) ) )
        | ? [TreeList3: list_VEBT_VEBT,N8: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),A22,TreeList3),Summary3) )
            & ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),N8) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary3),N8)
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N8),N8) )
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8))
               => ( ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),X_12)
                <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary3),I4) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi3 != Ma3 )
             => ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N8) = I4 )
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N8)) )
                    & ! [X: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N8) = I4 )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N8)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma3) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N8: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),A22,TreeList3),Summary3) )
            & ! [X: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X),N8) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary3),aa_nat_nat(suc,N8))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N8)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N8),aa_nat_nat(suc,N8)) )
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N8)))
               => ( ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),X_12)
                <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary3),I4) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi3 != Ma3 )
             => ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N8)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N8) = I4 )
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N8)) )
                    & ! [X: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N8) = I4 )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList3,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N8)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma3) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_1568_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
     => ( ! [Uu2: $o,Uv: $o] : Xb != vEBT_Leaf((Uu2),(Uv))
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,Va3),Vb2)
               => ( ( Xaa = Mi2 )
                  | ( Xaa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)
                 => ( ( Xaa = Mi2 )
                    | ( Xaa = Ma2 )
                    | $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_1569_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv: $o] : Xb = vEBT_Leaf((Uu2),(Uv))
         => (Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)
           => (Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,Va3),Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_1570_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
     => ( ! [A: $o,B: $o] :
            ( ( Xb = vEBT_Leaf((A),(B)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A),
                $ite(Xaa = one_one_nat,(B),$false) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv),Uw)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2)
               => $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_1571_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
     => ( ! [A: $o,B: $o] :
            ( ( Xb = vEBT_Leaf((A),(B)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A),
                  $ite(Xaa = one_one_nat,(B),$false) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2)
             => ~ $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_1572_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
      <=> (Y) )
     => ( ! [A: $o,B: $o] :
            ( ( Xb = vEBT_Leaf((A),(B)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A),
                    $ite(Xaa = one_one_nat,(B),$false) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv),Uw)
           => (Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2)
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_1573_even__odd__cases,axiom,
    ! [Xb: nat] :
      ( ! [N3: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),N3)
     => ~ ! [N3: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_nat_nat(suc,N3)) ) ).

% even_odd_cases
tff(fact_1574_add__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_nat_option_nat(some_nat,Xb)),aa_nat_option_nat(some_nat,Y)) = aa_nat_option_nat(some_nat,Z) ) ) ).

% add_shift
tff(fact_1575_both__member__options__def,axiom,
    ! [Tb: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Tb),Xb)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Tb),Xb)
        | aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Tb),Xb) ) ) ).

% both_member_options_def
tff(fact_1576_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_1577_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),N)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tree),Xb)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Tree),Xb)
          | aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Tree),Xb) ) ) ) ).

% member_valid_both_member_options
tff(fact_1578_pow__sum,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2) ).

% pow_sum
tff(fact_1579_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ).

% high_bound_aux
tff(fact_1580_add__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(zero_z3563351764282998399l_num1,A2) = A2 ).

% add_0
tff(fact_1581_add__0,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A2) = A2 ).

% add_0
tff(fact_1582_add__0,axiom,
    ! [A2: rat] : plus_plus_rat(zero_zero_rat,A2) = A2 ).

% add_0
tff(fact_1583_add__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A2) = A2 ).

% add_0
tff(fact_1584_add__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A2) = A2 ).

% add_0
tff(fact_1585_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) )
    <=> ( ( Xb = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_1586_add__eq__0__iff__both__eq__0,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
    <=> ( ( Xb = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_1587_add__cancel__right__right,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( A2 = plus_p361126936061061375l_num1(A2,B2) )
    <=> ( B2 = zero_z3563351764282998399l_num1 ) ) ).

% add_cancel_right_right
tff(fact_1588_add__cancel__right__right,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_1589_add__cancel__right__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = plus_plus_rat(A2,B2) )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_1590_add__cancel__right__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_1591_add__cancel__right__right,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_1592_add__cancel__right__left,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( A2 = plus_p361126936061061375l_num1(B2,A2) )
    <=> ( B2 = zero_z3563351764282998399l_num1 ) ) ).

% add_cancel_right_left
tff(fact_1593_add__cancel__right__left,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2) )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_1594_add__cancel__right__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = plus_plus_rat(B2,A2) )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_1595_add__cancel__right__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_1596_add__cancel__right__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2) )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_1597_add__cancel__left__right,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1(A2,B2) = A2 )
    <=> ( B2 = zero_z3563351764282998399l_num1 ) ) ).

% add_cancel_left_right
tff(fact_1598_add__cancel__left__right,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = A2 )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_1599_add__cancel__left__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( plus_plus_rat(A2,B2) = A2 )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_1600_add__cancel__left__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) = A2 )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_1601_add__cancel__left__right,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = A2 )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_1602_add__cancel__left__left,axiom,
    ! [B2: word_N3645301735248828278l_num1,A2: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1(B2,A2) = A2 )
    <=> ( B2 = zero_z3563351764282998399l_num1 ) ) ).

% add_cancel_left_left
tff(fact_1603_add__cancel__left__left,axiom,
    ! [B2: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2) = A2 )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_1604_add__cancel__left__left,axiom,
    ! [B2: rat,A2: rat] :
      ( ( plus_plus_rat(B2,A2) = A2 )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_1605_add__cancel__left__left,axiom,
    ! [B2: nat,A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) = A2 )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_1606_add__cancel__left__left,axiom,
    ! [B2: int,A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2) = A2 )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_1607_double__zero__sym,axiom,
    ! [A2: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2) )
    <=> ( A2 = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_1608_double__zero__sym,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat = plus_plus_rat(A2,A2) )
    <=> ( A2 = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_1609_double__zero__sym,axiom,
    ! [A2: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2) )
    <=> ( A2 = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_1610_add_Oright__neutral,axiom,
    ! [A2: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(A2,zero_z3563351764282998399l_num1) = A2 ).

% add.right_neutral
tff(fact_1611_add_Oright__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),zero_zero_real) = A2 ).

% add.right_neutral
tff(fact_1612_add_Oright__neutral,axiom,
    ! [A2: rat] : plus_plus_rat(A2,zero_zero_rat) = A2 ).

% add.right_neutral
tff(fact_1613_add_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),zero_zero_nat) = A2 ).

% add.right_neutral
tff(fact_1614_add_Oright__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),zero_zero_int) = A2 ).

% add.right_neutral
tff(fact_1615_double__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_1616_double__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat(A2,A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_1617_double__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_1618_add__le__cancel__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% add_le_cancel_right
tff(fact_1619_add__le__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A2,C)),plus_plus_rat(B2,C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% add_le_cancel_right
tff(fact_1620_add__le__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ).

% add_le_cancel_right
tff(fact_1621_add__le__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% add_le_cancel_right
tff(fact_1622_add__le__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% add_le_cancel_left
tff(fact_1623_add__le__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(C,A2)),plus_plus_rat(C,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% add_le_cancel_left
tff(fact_1624_add__le__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ).

% add_le_cancel_left
tff(fact_1625_add__le__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% add_le_cancel_left
tff(fact_1626_add__less__cancel__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% add_less_cancel_right
tff(fact_1627_add__less__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A2,C)),plus_plus_rat(B2,C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% add_less_cancel_right
tff(fact_1628_add__less__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% add_less_cancel_right
tff(fact_1629_add__less__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% add_less_cancel_right
tff(fact_1630_add__less__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% add_less_cancel_left
tff(fact_1631_add__less__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(C,A2)),plus_plus_rat(C,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% add_less_cancel_left
tff(fact_1632_add__less__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% add_less_cancel_left
tff(fact_1633_add__less__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% add_less_cancel_left
tff(fact_1634_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : plus_p361126936061061375l_num1(numera7442385471795722001l_num1(M),numera7442385471795722001l_num1(N)) = numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_1635_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(M)),numeral_numeral_real(N)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_1636_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : plus_plus_rat(numeral_numeral_rat(M),numeral_numeral_rat(N)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_1637_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_1638_numeral__plus__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% numeral_plus_numeral
tff(fact_1639_add__numeral__left,axiom,
    ! [V: num,W: num,Z: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(numera7442385471795722001l_num1(V),plus_p361126936061061375l_num1(numera7442385471795722001l_num1(W),Z)) = plus_p361126936061061375l_num1(numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)),Z) ).

% add_numeral_left
tff(fact_1640_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_1641_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(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)),Z) ).

% add_numeral_left
tff(fact_1642_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_1643_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Z) ).

% add_numeral_left
tff(fact_1644_add__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% add_Suc_right
tff(fact_1645_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).

% Nat.add_0_right
tff(fact_1646_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_1647_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% nat_add_left_cancel_less
tff(fact_1648_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I,set_or4548717258645045905et_nat(L,U))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(L),I)
        & aa_set_nat_o(ord_less_eq_set_nat(I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_1649_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( aa_set_rat_o(member_rat(I),set_or633870826150836451st_rat(L,U))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),I)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_1650_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( aa_set_num_o(member_num(I),set_or7049704709247886629st_num(L,U))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),I)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_1651_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_1652_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( aa_set_int_o(member_int(I),set_or1266510415728281911st_int(L,U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_1653_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( aa_set_real_o(member_real(I),set_or1222579329274155063t_real(L,U))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),I)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_1654_Icc__eq__Icc,axiom,
    ! [L: set_nat,H: set_nat,L3: set_nat,H2: set_nat] :
      ( ( set_or4548717258645045905et_nat(L,H) = set_or4548717258645045905et_nat(L3,H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ aa_set_nat_o(ord_less_eq_set_nat(L),H)
          & ~ aa_set_nat_o(ord_less_eq_set_nat(L3),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_1655_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L3: rat,H2: rat] :
      ( ( set_or633870826150836451st_rat(L,H) = set_or633870826150836451st_rat(L3,H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),H)
          & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L3),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_1656_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L3: num,H2: num] :
      ( ( set_or7049704709247886629st_num(L,H) = set_or7049704709247886629st_num(L3,H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),H)
          & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L3),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_1657_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L3: nat,H2: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L),H) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,L3),H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
          & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L3),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_1658_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L3: int,H2: int] :
      ( ( set_or1266510415728281911st_int(L,H) = set_or1266510415728281911st_int(L3,H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),H)
          & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L3),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_1659_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L3: real,H2: real] :
      ( ( set_or1222579329274155063t_real(L,H) = set_or1222579329274155063t_real(L3,H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),H)
          & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_1660_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% nat_add_left_cancel_le
tff(fact_1661_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_1662_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1663_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),plus_plus_rat(A2,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1664_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1665_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1666_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A2,A2)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1667_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1668_le__add__same__cancel2,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) ).

% le_add_same_cancel2
tff(fact_1669_le__add__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),plus_plus_rat(B2,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) ).

% le_add_same_cancel2
tff(fact_1670_le__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2) ) ).

% le_add_same_cancel2
tff(fact_1671_le__add__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) ) ).

% le_add_same_cancel2
tff(fact_1672_le__add__same__cancel1,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) ).

% le_add_same_cancel1
tff(fact_1673_le__add__same__cancel1,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),plus_plus_rat(A2,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) ).

% le_add_same_cancel1
tff(fact_1674_le__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2) ) ).

% le_add_same_cancel1
tff(fact_1675_le__add__same__cancel1,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) ) ).

% le_add_same_cancel1
tff(fact_1676_add__le__same__cancel2,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_1677_add__le__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A2,B2)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_1678_add__le__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_1679_add__le__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_1680_add__le__same__cancel1,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_1681_add__le__same__cancel1,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(B2,A2)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_1682_add__le__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_1683_add__le__same__cancel1,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_1684_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1685_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),plus_plus_rat(A2,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1686_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1687_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1688_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A2,A2)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1689_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),A2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1690_less__add__same__cancel2,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ).

% less_add_same_cancel2
tff(fact_1691_less__add__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),plus_plus_rat(B2,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ).

% less_add_same_cancel2
tff(fact_1692_less__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2) ) ).

% less_add_same_cancel2
tff(fact_1693_less__add__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ).

% less_add_same_cancel2
tff(fact_1694_less__add__same__cancel1,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ).

% less_add_same_cancel1
tff(fact_1695_less__add__same__cancel1,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),plus_plus_rat(A2,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ).

% less_add_same_cancel1
tff(fact_1696_less__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2) ) ).

% less_add_same_cancel1
tff(fact_1697_less__add__same__cancel1,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ).

% less_add_same_cancel1
tff(fact_1698_add__less__same__cancel2,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_1699_add__less__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A2,B2)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_1700_add__less__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_1701_add__less__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_1702_add__less__same__cancel1,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_1703_add__less__same__cancel1,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(B2,A2)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_1704_add__less__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_1705_add__less__same__cancel1,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_1706_diff__add__zero,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) = zero_zero_nat ).

% diff_add_zero
tff(fact_1707_le__add__diff__inverse,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(minus_minus_real(A2),B2)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_1708_le__add__diff__inverse,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( plus_plus_rat(B2,minus_minus_rat(A2,B2)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_1709_le__add__diff__inverse,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_1710_le__add__diff__inverse,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_1711_le__add__diff__inverse2,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(minus_minus_real(A2),B2)),B2) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_1712_le__add__diff__inverse2,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( plus_plus_rat(minus_minus_rat(A2,B2),B2) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_1713_le__add__diff__inverse2,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)),B2) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_1714_le__add__diff__inverse2,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),B2) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_1715_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% add_gr_0
tff(fact_1716_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_1717_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1718_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1719_one__plus__numeral,axiom,
    ! [N: num] : plus_plus_uint32(one_one_uint32,numera9087168376688890119uint32(N)) = numera9087168376688890119uint32(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_1720_one__plus__numeral,axiom,
    ! [N: num] : plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,numera7442385471795722001l_num1(N)) = numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_1721_one__plus__numeral,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(N)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_1722_one__plus__numeral,axiom,
    ! [N: num] : plus_plus_rat(one_one_rat,numeral_numeral_rat(N)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_1723_one__plus__numeral,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_1724_one__plus__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% one_plus_numeral
tff(fact_1725_numeral__plus__one,axiom,
    ! [N: num] : plus_plus_uint32(numera9087168376688890119uint32(N),one_one_uint32) = numera9087168376688890119uint32(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_1726_numeral__plus__one,axiom,
    ! [N: num] : plus_p361126936061061375l_num1(numera7442385471795722001l_num1(N),one_on7727431528512463931l_num1) = numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_1727_numeral__plus__one,axiom,
    ! [N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),one_one_real) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_1728_numeral__plus__one,axiom,
    ! [N: num] : plus_plus_rat(numeral_numeral_rat(N),one_one_rat) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_1729_numeral__plus__one,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_1730_numeral__plus__one,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% numeral_plus_one
tff(fact_1731_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_1732_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_1733_one__add__one,axiom,
    plus_plus_uint32(one_one_uint32,one_one_uint32) = numera9087168376688890119uint32(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1734_one__add__one,axiom,
    plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,one_on7727431528512463931l_num1) = numera7442385471795722001l_num1(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1735_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1736_one__add__one,axiom,
    plus_plus_rat(one_one_rat,one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1737_one__add__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1738_one__add__one,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_1739_add__2__eq__Suc_H,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc'
tff(fact_1740_add__2__eq__Suc,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = aa_nat_nat(suc,aa_nat_nat(suc,N)) ).

% add_2_eq_Suc
tff(fact_1741_add__self__div__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = M ).

% add_self_div_2
tff(fact_1742_sum__power2__eq__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1743_sum__power2__eq__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1744_sum__power2__eq__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1745_sum__power2__eq__zero__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger )
    <=> ( ( Xb = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1746_is__num__normalize_I1_J,axiom,
    ! [A2: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C)) ).

% is_num_normalize(1)
tff(fact_1747_is__num__normalize_I1_J,axiom,
    ! [A2: rat,B2: rat,C: rat] : plus_plus_rat(plus_plus_rat(A2,B2),C) = plus_plus_rat(A2,plus_plus_rat(B2,C)) ).

% is_num_normalize(1)
tff(fact_1748_is__num__normalize_I1_J,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ).

% is_num_normalize(1)
tff(fact_1749_add_Ogroup__left__neutral,axiom,
    ! [A2: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(zero_z3563351764282998399l_num1,A2) = A2 ).

% add.group_left_neutral
tff(fact_1750_add_Ogroup__left__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A2) = A2 ).

% add.group_left_neutral
tff(fact_1751_add_Ogroup__left__neutral,axiom,
    ! [A2: rat] : plus_plus_rat(zero_zero_rat,A2) = A2 ).

% add.group_left_neutral
tff(fact_1752_add_Ogroup__left__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A2) = A2 ).

% add.group_left_neutral
tff(fact_1753_add_Ocomm__neutral,axiom,
    ! [A2: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(A2,zero_z3563351764282998399l_num1) = A2 ).

% add.comm_neutral
tff(fact_1754_add_Ocomm__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),zero_zero_real) = A2 ).

% add.comm_neutral
tff(fact_1755_add_Ocomm__neutral,axiom,
    ! [A2: rat] : plus_plus_rat(A2,zero_zero_rat) = A2 ).

% add.comm_neutral
tff(fact_1756_add_Ocomm__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),zero_zero_nat) = A2 ).

% add.comm_neutral
tff(fact_1757_add_Ocomm__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),zero_zero_int) = A2 ).

% add.comm_neutral
tff(fact_1758_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(zero_z3563351764282998399l_num1,A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_1759_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_1760_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: rat] : plus_plus_rat(zero_zero_rat,A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_1761_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_1762_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_1763_add__le__imp__le__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% add_le_imp_le_right
tff(fact_1764_add__le__imp__le__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A2,C)),plus_plus_rat(B2,C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% add_le_imp_le_right
tff(fact_1765_add__le__imp__le__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ).

% add_le_imp_le_right
tff(fact_1766_add__le__imp__le__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% add_le_imp_le_right
tff(fact_1767_add__le__imp__le__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B2))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% add_le_imp_le_left
tff(fact_1768_add__le__imp__le__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(C,A2)),plus_plus_rat(C,B2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% add_le_imp_le_left
tff(fact_1769_add__le__imp__le__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ).

% add_le_imp_le_left
tff(fact_1770_add__le__imp__le__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% add_le_imp_le_left
tff(fact_1771_le__iff__add,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
    <=> ? [C3: nat] : B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C3) ) ).

% le_iff_add
tff(fact_1772_add__right__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C)) ) ).

% add_right_mono
tff(fact_1773_add__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A2,C)),plus_plus_rat(B2,C)) ) ).

% add_right_mono
tff(fact_1774_add__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)) ) ).

% add_right_mono
tff(fact_1775_add__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ) ).

% add_right_mono
tff(fact_1776_less__eqE,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ~ ! [C2: nat] : B2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2) ) ).

% less_eqE
tff(fact_1777_add__left__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B2)) ) ).

% add_left_mono
tff(fact_1778_add__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(C,A2)),plus_plus_rat(C,B2)) ) ).

% add_left_mono
tff(fact_1779_add__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)) ) ).

% add_left_mono
tff(fact_1780_add__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)) ) ).

% add_left_mono
tff(fact_1781_add__mono,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D)) ) ) ).

% add_mono
tff(fact_1782_add__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A2,C)),plus_plus_rat(B2,D)) ) ) ).

% add_mono
tff(fact_1783_add__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D)) ) ) ).

% add_mono
tff(fact_1784_add__mono,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D)) ) ) ).

% add_mono
tff(fact_1785_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1786_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(I,K)),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1787_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1788_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1789_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1790_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(I,K)),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1791_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1792_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1793_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1794_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(I,K)),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1795_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1796_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1797_add__less__imp__less__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% add_less_imp_less_right
tff(fact_1798_add__less__imp__less__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A2,C)),plus_plus_rat(B2,C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% add_less_imp_less_right
tff(fact_1799_add__less__imp__less__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% add_less_imp_less_right
tff(fact_1800_add__less__imp__less__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% add_less_imp_less_right
tff(fact_1801_add__less__imp__less__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B2))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% add_less_imp_less_left
tff(fact_1802_add__less__imp__less__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(C,A2)),plus_plus_rat(C,B2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% add_less_imp_less_left
tff(fact_1803_add__less__imp__less__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% add_less_imp_less_left
tff(fact_1804_add__less__imp__less__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% add_less_imp_less_left
tff(fact_1805_add__strict__right__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C)) ) ).

% add_strict_right_mono
tff(fact_1806_add__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A2,C)),plus_plus_rat(B2,C)) ) ).

% add_strict_right_mono
tff(fact_1807_add__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)) ) ).

% add_strict_right_mono
tff(fact_1808_add__strict__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ) ).

% add_strict_right_mono
tff(fact_1809_add__strict__left__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B2)) ) ).

% add_strict_left_mono
tff(fact_1810_add__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(C,A2)),plus_plus_rat(C,B2)) ) ).

% add_strict_left_mono
tff(fact_1811_add__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)) ) ).

% add_strict_left_mono
tff(fact_1812_add__strict__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)) ) ).

% add_strict_left_mono
tff(fact_1813_add__strict__mono,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D)) ) ) ).

% add_strict_mono
tff(fact_1814_add__strict__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A2,C)),plus_plus_rat(B2,D)) ) ) ).

% add_strict_mono
tff(fact_1815_add__strict__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D)) ) ) ).

% add_strict_mono
tff(fact_1816_add__strict__mono,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D)) ) ) ).

% add_strict_mono
tff(fact_1817_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1818_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(I,K)),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1819_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1820_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1821_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1822_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(I,K)),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1823_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1824_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1825_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1826_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(I,K)),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1827_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1828_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1829_add__divide__distrib,axiom,
    ! [A2: real,B2: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(A2),C)),aa_real_real(divide_divide_real(B2),C)) ).

% add_divide_distrib
tff(fact_1830_add__divide__distrib,axiom,
    ! [A2: rat,B2: rat,C: rat] : divide_divide_rat(plus_plus_rat(A2,B2),C) = plus_plus_rat(divide_divide_rat(A2,C),divide_divide_rat(B2,C)) ).

% add_divide_distrib
tff(fact_1831_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,N)) ).

% add_Suc_shift
tff(fact_1832_add__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% add_Suc
tff(fact_1833_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A2: nat] :
      ( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A2) )
     => ( aa_nat_nat(suc,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A2)) ) ) ).

% nat_arith.suc1
tff(fact_1834_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_1835_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N ).

% plus_nat.add_0
tff(fact_1836_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% less_add_eq_less
tff(fact_1837_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_less_add2
tff(fact_1838_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_less_add1
tff(fact_1839_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_less_mono1
tff(fact_1840_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I) ).

% not_add_less2
tff(fact_1841_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I) ).

% not_add_less1
tff(fact_1842_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_1843_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% add_lessD1
tff(fact_1844_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
    <=> ? [K3: nat] : N = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3) ) ).

% nat_le_iff_add
tff(fact_1845_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_le_add2
tff(fact_1846_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_le_add1
tff(fact_1847_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_le_mono1
tff(fact_1848_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_1849_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
     => ? [N3: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N3) ) ).

% le_Suc_ex
tff(fact_1850_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) ) ).

% add_leD2
tff(fact_1851_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% add_leD1
tff(fact_1852_le__add2,axiom,
    ! [N: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% le_add2
tff(fact_1853_le__add1,axiom,
    ! [N: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) ).

% le_add1
tff(fact_1854_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),N)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) ) ) ).

% add_leE
tff(fact_1855_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = M ).

% diff_add_inverse2
tff(fact_1856_diff__add__inverse,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),N) = M ).

% diff_add_inverse
tff(fact_1857_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% diff_cancel2
tff(fact_1858_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N) ).

% Nat.diff_cancel
tff(fact_1859_add__nonpos__eq__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1860_add__nonpos__eq__0__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => ( ( plus_plus_rat(Xb,Y) = zero_zero_rat )
        <=> ( ( Xb = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1861_add__nonpos__eq__0__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1862_add__nonpos__eq__0__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1863_add__nonneg__eq__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1864_add__nonneg__eq__0__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( plus_plus_rat(Xb,Y) = zero_zero_rat )
        <=> ( ( Xb = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1865_add__nonneg__eq__0__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1866_add__nonneg__eq__0__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1867_add__nonpos__nonpos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_1868_add__nonpos__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A2,B2)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_1869_add__nonpos__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_1870_add__nonpos__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_1871_add__nonneg__nonneg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) ) ) ).

% add_nonneg_nonneg
tff(fact_1872_add__nonneg__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),plus_plus_rat(A2,B2)) ) ) ).

% add_nonneg_nonneg
tff(fact_1873_add__nonneg__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ) ) ).

% add_nonneg_nonneg
tff(fact_1874_add__nonneg__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)) ) ) ).

% add_nonneg_nonneg
tff(fact_1875_add__increasing2,axiom,
    ! [C: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)) ) ) ).

% add_increasing2
tff(fact_1876_add__increasing2,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),plus_plus_rat(A2,C)) ) ) ).

% add_increasing2
tff(fact_1877_add__increasing2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)) ) ) ).

% add_increasing2
tff(fact_1878_add__increasing2,axiom,
    ! [C: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)) ) ) ).

% add_increasing2
tff(fact_1879_add__decreasing2,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),B2) ) ) ).

% add_decreasing2
tff(fact_1880_add__decreasing2,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A2,C)),B2) ) ) ).

% add_decreasing2
tff(fact_1881_add__decreasing2,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),B2) ) ) ).

% add_decreasing2
tff(fact_1882_add__decreasing2,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),B2) ) ) ).

% add_decreasing2
tff(fact_1883_add__increasing,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)) ) ) ).

% add_increasing
tff(fact_1884_add__increasing,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),plus_plus_rat(A2,C)) ) ) ).

% add_increasing
tff(fact_1885_add__increasing,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)) ) ) ).

% add_increasing
tff(fact_1886_add__increasing,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)) ) ) ).

% add_increasing
tff(fact_1887_add__decreasing,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),B2) ) ) ).

% add_decreasing
tff(fact_1888_add__decreasing,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A2,C)),B2) ) ) ).

% add_decreasing
tff(fact_1889_add__decreasing,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),B2) ) ) ).

% add_decreasing
tff(fact_1890_add__decreasing,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),B2) ) ) ).

% add_decreasing
tff(fact_1891_pos__add__strict,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)) ) ) ).

% pos_add_strict
tff(fact_1892_pos__add__strict,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),plus_plus_rat(A2,C)) ) ) ).

% pos_add_strict
tff(fact_1893_pos__add__strict,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)) ) ) ).

% pos_add_strict
tff(fact_1894_pos__add__strict,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)) ) ) ).

% pos_add_strict
tff(fact_1895_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ~ ! [C2: nat] :
            ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1896_add__pos__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) ) ) ).

% add_pos_pos
tff(fact_1897_add__pos__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),plus_plus_rat(A2,B2)) ) ) ).

% add_pos_pos
tff(fact_1898_add__pos__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ) ) ).

% add_pos_pos
tff(fact_1899_add__pos__pos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)) ) ) ).

% add_pos_pos
tff(fact_1900_add__neg__neg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_1901_add__neg__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A2,B2)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_1902_add__neg__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_1903_add__neg__neg,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_1904_add__less__zeroD,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_1905_add__less__zeroD,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(Xb,Y)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_1906_add__less__zeroD,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_1907_add__less__le__mono,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D)) ) ) ).

% add_less_le_mono
tff(fact_1908_add__less__le__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A2,C)),plus_plus_rat(B2,D)) ) ) ).

% add_less_le_mono
tff(fact_1909_add__less__le__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D)) ) ) ).

% add_less_le_mono
tff(fact_1910_add__less__le__mono,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D)) ) ) ).

% add_less_le_mono
tff(fact_1911_add__le__less__mono,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D)) ) ) ).

% add_le_less_mono
tff(fact_1912_add__le__less__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A2,C)),plus_plus_rat(B2,D)) ) ) ).

% add_le_less_mono
tff(fact_1913_add__le__less__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D)) ) ) ).

% add_le_less_mono
tff(fact_1914_add__le__less__mono,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D)) ) ) ).

% add_le_less_mono
tff(fact_1915_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1916_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(I,K)),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1917_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1918_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1919_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1920_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(I,K)),plus_plus_rat(J,L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1921_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1922_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1923_numeral__Bit0,axiom,
    ! [N: num] : numera7442385471795722001l_num1(aa_num_num(bit0,N)) = plus_p361126936061061375l_num1(numera7442385471795722001l_num1(N),numera7442385471795722001l_num1(N)) ).

% numeral_Bit0
tff(fact_1924_numeral__Bit0,axiom,
    ! [N: num] : numeral_numeral_real(aa_num_num(bit0,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),numeral_numeral_real(N)) ).

% numeral_Bit0
tff(fact_1925_numeral__Bit0,axiom,
    ! [N: num] : numeral_numeral_rat(aa_num_num(bit0,N)) = plus_plus_rat(numeral_numeral_rat(N),numeral_numeral_rat(N)) ).

% numeral_Bit0
tff(fact_1926_numeral__Bit0,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_Bit0
tff(fact_1927_numeral__Bit0,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N)) ).

% numeral_Bit0
tff(fact_1928_add__mono1,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),one_one_real)) ) ).

% add_mono1
tff(fact_1929_add__mono1,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A2,one_one_rat)),plus_plus_rat(B2,one_one_rat)) ) ).

% add_mono1
tff(fact_1930_add__mono1,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),one_one_nat)) ) ).

% add_mono1
tff(fact_1931_add__mono1,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),one_one_int)) ) ).

% add_mono1
tff(fact_1932_less__add__one,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),one_one_real)) ).

% less_add_one
tff(fact_1933_less__add__one,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),plus_plus_rat(A2,one_one_rat)) ).

% less_add_one
tff(fact_1934_less__add__one,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)) ).

% less_add_one
tff(fact_1935_less__add__one,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int)) ).

% less_add_one
tff(fact_1936_one__plus__numeral__commute,axiom,
    ! [Xb: num] : plus_plus_uint32(one_one_uint32,numera9087168376688890119uint32(Xb)) = plus_plus_uint32(numera9087168376688890119uint32(Xb),one_one_uint32) ).

% one_plus_numeral_commute
tff(fact_1937_one__plus__numeral__commute,axiom,
    ! [Xb: num] : plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,numera7442385471795722001l_num1(Xb)) = plus_p361126936061061375l_num1(numera7442385471795722001l_num1(Xb),one_on7727431528512463931l_num1) ).

% one_plus_numeral_commute
tff(fact_1938_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),numeral_numeral_real(Xb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(Xb)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_1939_one__plus__numeral__commute,axiom,
    ! [Xb: num] : plus_plus_rat(one_one_rat,numeral_numeral_rat(Xb)) = plus_plus_rat(numeral_numeral_rat(Xb),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_1940_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_1941_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Xb)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_1942_diff__le__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A2),B2)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B2)) ) ).

% diff_le_eq
tff(fact_1943_diff__le__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A2,B2)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),plus_plus_rat(C,B2)) ) ).

% diff_le_eq
tff(fact_1944_diff__le__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)) ) ).

% diff_le_eq
tff(fact_1945_le__diff__eq,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(minus_minus_real(C),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C) ) ).

% le_diff_eq
tff(fact_1946_le__diff__eq,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),minus_minus_rat(C,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(A2,B2)),C) ) ).

% le_diff_eq
tff(fact_1947_le__diff__eq,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) ) ).

% le_diff_eq
tff(fact_1948_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)),A2) = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_1949_le__add__diff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)),A2)) ) ).

% le_add_diff
tff(fact_1950_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),B2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1951_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)),A2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1952_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1953_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)),A2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1954_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1955_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2)),B2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1956_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)) = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1957_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2) = C )
        <=> ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A2) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1958_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),aa_real_real(minus_minus_real(N),K)) ) ).

% add_le_imp_le_diff
tff(fact_1959_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(I,K)),N)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),minus_minus_rat(N,K)) ) ).

% add_le_imp_le_diff
tff(fact_1960_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ).

% add_le_imp_le_diff
tff(fact_1961_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K)) ) ).

% add_le_imp_le_diff
tff(fact_1962_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N: real,J: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),N)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,N),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(N),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1963_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(I,K)),N)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,N),plus_plus_rat(J,K))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(I,K)),N)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,N),plus_plus_rat(J,K))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(N,K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1964_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),N)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1965_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),N)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1966_less__diff__eq,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(minus_minus_real(C),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C) ) ).

% less_diff_eq
tff(fact_1967_less__diff__eq,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),minus_minus_rat(C,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A2,B2)),C) ) ).

% less_diff_eq
tff(fact_1968_less__diff__eq,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) ) ).

% less_diff_eq
tff(fact_1969_diff__less__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A2),B2)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B2)) ) ).

% diff_less_eq
tff(fact_1970_diff__less__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A2,B2)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),plus_plus_rat(C,B2)) ) ).

% diff_less_eq
tff(fact_1971_diff__less__eq,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)) ) ).

% diff_less_eq
tff(fact_1972_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: real,B2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(minus_minus_real(A2),B2)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1973_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: rat,B2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( plus_plus_rat(B2,minus_minus_rat(A2,B2)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1974_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1975_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: int,B2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1976_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_1977_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_1978_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ~ ! [Q3: nat] : N != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ) ).

% less_natE
tff(fact_1979_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M))) ).

% less_add_Suc1
tff(fact_1980_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I))) ).

% less_add_Suc2
tff(fact_1981_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
    <=> ? [K3: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_iff_Suc_add
tff(fact_1982_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ? [K2: nat] : N = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).

% less_imp_Suc_add
tff(fact_1983_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_1984_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),N) ).

% Suc_eq_plus1_left
tff(fact_1985_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_1986_Suc__eq__plus1,axiom,
    ! [N: nat] : aa_nat_nat(suc,N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat) ).

% Suc_eq_plus1
tff(fact_1987_mono__nat__linear__lb,axiom,
    ! [Fa: fun_nat_nat,M: nat,K: nat] :
      ( ! [M3: nat,N3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(Fa,M3)),aa_nat_nat(Fa,N3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Fa,M)),K)),aa_nat_nat(Fa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))) ) ).

% mono_nat_linear_lb
tff(fact_1988_diff__add__0,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_1989_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ).

% less_diff_conv
tff(fact_1990_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_1991_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ).

% le_diff_conv
tff(fact_1992_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_1993_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1994_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1995_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1996_numeral__code_I2_J,axiom,
    ! [N: num] :
      numera7442385471795722001l_num1(aa_num_num(bit0,N)) = $let(
        m: word_N3645301735248828278l_num1,
        m:= numera7442385471795722001l_num1(N),
        plus_p361126936061061375l_num1(m,m) ) ).

% numeral_code(2)
tff(fact_1997_numeral__code_I2_J,axiom,
    ! [N: num] :
      numeral_numeral_real(aa_num_num(bit0,N)) = $let(
        m: real,
        m:= numeral_numeral_real(N),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m),m) ) ).

% numeral_code(2)
tff(fact_1998_numeral__code_I2_J,axiom,
    ! [N: num] :
      numeral_numeral_rat(aa_num_num(bit0,N)) = $let(
        m: rat,
        m:= numeral_numeral_rat(N),
        plus_plus_rat(m,m) ) ).

% numeral_code(2)
tff(fact_1999_numeral__code_I2_J,axiom,
    ! [N: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = $let(
        m: nat,
        m:= aa_num_nat(numeral_numeral_nat,N),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m) ) ).

% numeral_code(2)
tff(fact_2000_numeral__code_I2_J,axiom,
    ! [N: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,N),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m),m) ) ).

% numeral_code(2)
tff(fact_2001_VEBT_Odisc__eq__case_I1_J,axiom,
    ! [VEBT: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_is_Node,VEBT)
    <=> aa_VEBT_VEBT_o(vEBT_case_VEBT_o(aTP_Lamm_iq,aTP_Lamm_ir),VEBT) ) ).

% VEBT.disc_eq_case(1)
tff(fact_2002_add__strict__increasing2,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)) ) ) ).

% add_strict_increasing2
tff(fact_2003_add__strict__increasing2,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),plus_plus_rat(A2,C)) ) ) ).

% add_strict_increasing2
tff(fact_2004_add__strict__increasing2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)) ) ) ).

% add_strict_increasing2
tff(fact_2005_add__strict__increasing2,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)) ) ) ).

% add_strict_increasing2
tff(fact_2006_add__strict__increasing,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)) ) ) ).

% add_strict_increasing
tff(fact_2007_add__strict__increasing,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),plus_plus_rat(A2,C)) ) ) ).

% add_strict_increasing
tff(fact_2008_add__strict__increasing,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C)) ) ) ).

% add_strict_increasing
tff(fact_2009_add__strict__increasing,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C)) ) ) ).

% add_strict_increasing
tff(fact_2010_add__pos__nonneg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) ) ) ).

% add_pos_nonneg
tff(fact_2011_add__pos__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),plus_plus_rat(A2,B2)) ) ) ).

% add_pos_nonneg
tff(fact_2012_add__pos__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ) ) ).

% add_pos_nonneg
tff(fact_2013_add__pos__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)) ) ) ).

% add_pos_nonneg
tff(fact_2014_add__nonpos__neg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_2015_add__nonpos__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A2,B2)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_2016_add__nonpos__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_2017_add__nonpos__neg,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_2018_add__nonneg__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) ) ) ).

% add_nonneg_pos
tff(fact_2019_add__nonneg__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),plus_plus_rat(A2,B2)) ) ) ).

% add_nonneg_pos
tff(fact_2020_add__nonneg__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ) ) ).

% add_nonneg_pos
tff(fact_2021_add__nonneg__pos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)) ) ) ).

% add_nonneg_pos
tff(fact_2022_add__neg__nonpos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_2023_add__neg__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(A2,B2)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_2024_add__neg__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_2025_add__neg__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_2026_field__le__epsilon,axiom,
    ! [Xb: real,Y: real] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% field_le_epsilon
tff(fact_2027_field__le__epsilon,axiom,
    ! [Xb: rat,Y: rat] :
      ( ! [E: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),plus_plus_rat(Y,E)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% field_le_epsilon
tff(fact_2028_zero__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)) ).

% zero_less_two
tff(fact_2029_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),plus_plus_rat(one_one_rat,one_one_rat)) ).

% zero_less_two
tff(fact_2030_zero__less__two,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)) ).

% zero_less_two
tff(fact_2031_zero__less__two,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)) ).

% zero_less_two
tff(fact_2032_discrete,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),B2) ) ).

% discrete
tff(fact_2033_discrete,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int)),B2) ) ).

% discrete
tff(fact_2034_div__add__self2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_2035_div__add__self2,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),one_one_int) ) ) ).

% div_add_self2
tff(fact_2036_div__add__self1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_2037_div__add__self1,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),one_one_int) ) ) ).

% div_add_self1
tff(fact_2038_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv2),Uw2)),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_2039_less__half__sum,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_2040_less__half__sum,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),divide_divide_rat(plus_plus_rat(A2,B2),plus_plus_rat(one_one_rat,one_one_rat))) ) ).

% less_half_sum
tff(fact_2041_gt__half__sum,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B2) ) ).

% gt_half_sum
tff(fact_2042_gt__half__sum,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(plus_plus_rat(A2,B2),plus_plus_rat(one_one_rat,one_one_rat))),B2) ) ).

% gt_half_sum
tff(fact_2043_nat__diff__split__asm,axiom,
    ! [P: fun_nat_o,A2: nat,B2: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
            & ~ aa_nat_o(P,zero_zero_nat) )
          | ? [D3: nat] :
              ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D3) )
              & ~ aa_nat_o(P,D3) ) ) ) ).

% nat_diff_split_asm
tff(fact_2044_nat__diff__split,axiom,
    ! [P: fun_nat_o,A2: nat,B2: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
         => aa_nat_o(P,zero_zero_nat) )
        & ! [D3: nat] :
            ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D3) )
           => aa_nat_o(P,D3) ) ) ) ).

% nat_diff_split
tff(fact_2045_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ) ).

% less_diff_conv2
tff(fact_2046_bounded__Max__nat,axiom,
    ! [P: fun_nat_o,Xb: nat,M8: nat] :
      ( aa_nat_o(P,Xb)
     => ( ! [X2: nat] :
            ( aa_nat_o(P,X2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),M8) )
       => ~ ! [M3: nat] :
              ( aa_nat_o(P,M3)
             => ~ ! [X3: nat] :
                    ( aa_nat_o(P,X3)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M3) ) ) ) ) ).

% bounded_Max_nat
tff(fact_2047_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [Xb: produc2554157576835007900T_VEBT] :
      ~ ! [F3: fun_na4687389303393967696T_VEBT,A: nat,B: nat,Acc: produc4813437837504472865T_VEBT] : Xb != produc381187579358852302T_VEBT(F3,produc5776433622635646767T_VEBT(A,aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(B),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_2048_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [Xb: produc8114805097830739791at_num] :
      ~ ! [F3: fun_nat_fun_num_num,A: nat,B: nat,Acc: num] : Xb != produc460779904311030791at_num(F3,produc1195630363706982562at_num(A,aa_num5026351239867140595at_num(product_Pair_nat_num(B),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_2049_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [Xb: produc9217583074051984497at_nat] :
      ~ ! [F3: fun_nat_fun_nat_nat,A: nat,B: nat,Acc: nat] : Xb != produc2818902965508115497at_nat(F3,produc487386426758144856at_nat(A,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_2050_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy)),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_2051_nat__1__add__1,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% nat_1_add_1
tff(fact_2052_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) = $ite(M = zero_zero_nat,N,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N))) ).

% add_eq_if
tff(fact_2053_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,vEBT_Leaf((A2),(B2))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (A2),
          $ite(Xb = one_one_nat,(B2),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_2054_num_Osize_I5_J,axiom,
    ! [X22: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_2055_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2056_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_z3563351764282998399l_num1 )
     => ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N) != zero_z3563351764282998399l_num1 ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2057_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2058_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2059_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2060_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_z3563351764282998399l_num1 )
     => ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),M) != zero_z3563351764282998399l_num1 ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2061_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2062_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2063_div__exp__eq,axiom,
    ! [A2: code_integer,M: nat,N: nat] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_2064_div__exp__eq,axiom,
    ! [A2: word_N3645301735248828278l_num1,M: nat,N: nat] : divide1791077408188789448l_num1(divide1791077408188789448l_num1(A2,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),M)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N)) = divide1791077408188789448l_num1(A2,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_2065_div__exp__eq,axiom,
    ! [A2: uint32,M: nat,N: nat] : aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),M))),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),N)) = aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_2066_div__exp__eq,axiom,
    ! [A2: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_2067_div__exp__eq,axiom,
    ! [A2: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ).

% div_exp_eq
tff(fact_2068_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va),Vb)),Xb)
    <=> ( ( Xb = Mi )
        | ( Xb = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_2069_sum__power2__le__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2070_sum__power2__le__zero__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_z3403309356797280102nteger)
    <=> ( ( Xb = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2071_sum__power2__le__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2072_sum__power2__le__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2073_sum__power2__ge__zero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_2074_sum__power2__ge__zero,axiom,
    ! [Xb: code_integer,Y: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_2075_sum__power2__ge__zero,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_2076_sum__power2__ge__zero,axiom,
    ! [Xb: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_2077_sum__power2__gt__zero__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2078_sum__power2__gt__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2079_sum__power2__gt__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2080_sum__power2__gt__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2081_not__sum__power2__lt__zero,axiom,
    ! [Xb: code_integer,Y: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_z3403309356797280102nteger) ).

% not_sum_power2_lt_zero
tff(fact_2082_not__sum__power2__lt__zero,axiom,
    ! [Xb: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real) ).

% not_sum_power2_lt_zero
tff(fact_2083_not__sum__power2__lt__zero,axiom,
    ! [Xb: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_2084_not__sum__power2__lt__zero,axiom,
    ! [Xb: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_2085_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
       => ? [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N3)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_2086_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K)
       => ? [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N3)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_2087_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xb: nat,N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_2088_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xb: nat,N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_2089_atLeastatMost__psubset__iff,axiom,
    ! [A2: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
      ( ord_less_set_set_nat(set_or4548717258645045905et_nat(A2,B2),set_or4548717258645045905et_nat(C,D))
    <=> ( ( ~ aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
          | ( aa_set_nat_o(ord_less_eq_set_nat(C),A2)
            & aa_set_nat_o(ord_less_eq_set_nat(B2),D)
            & ( aa_set_nat_o(ord_less_set_nat(C),A2)
              | aa_set_nat_o(ord_less_set_nat(B2),D) ) ) )
        & aa_set_nat_o(ord_less_eq_set_nat(C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2090_atLeastatMost__psubset__iff,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( aa_set_rat_o(ord_less_set_rat(set_or633870826150836451st_rat(A2,B2)),set_or633870826150836451st_rat(C,D))
    <=> ( ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
          | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),D)
            & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A2)
              | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),D) ) ) )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2091_atLeastatMost__psubset__iff,axiom,
    ! [A2: num,B2: num,C: num,D: num] :
      ( aa_set_num_o(ord_less_set_num(set_or7049704709247886629st_num(A2,B2)),set_or7049704709247886629st_num(C,D))
    <=> ( ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
          | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A2)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),D)
            & ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),A2)
              | aa_num_o(aa_num_fun_num_o(ord_less_num,B2),D) ) ) )
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2092_atLeastatMost__psubset__iff,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( aa_set_nat_o(ord_less_set_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A2),B2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,C),D))
    <=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),D)
            & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A2)
              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),D) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2093_atLeastatMost__psubset__iff,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_set_int(set_or1266510415728281911st_int(A2,B2)),set_or1266510415728281911st_int(C,D))
    <=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
          | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A2)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),D)
            & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A2)
              | aa_int_o(aa_int_fun_int_o(ord_less_int,B2),D) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2094_atLeastatMost__psubset__iff,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( aa_set_real_o(ord_less_set_real(set_or1222579329274155063t_real(A2,B2)),set_or1222579329274155063t_real(C,D))
    <=> ( ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
          | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),A2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),D)
            & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A2)
              | aa_real_o(aa_real_fun_real_o(ord_less_real,B2),D) ) ) )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2095_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X2),N) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = N )
           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X_1)
               => ( ! [X2: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList))
                     => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X2),X_1) )
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary2)),Deg2) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_2096_vebt__delete_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,N: nat] : vEBT_vebt_delete(vEBT_Leaf((A2),(B2)),aa_nat_nat(suc,aa_nat_nat(suc,N))) = vEBT_Leaf((A2),(B2)) ).

% vebt_delete.simps(3)
tff(fact_2097_vebt__delete_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_vebt_delete(vEBT_Leaf((A2),(B2)),zero_zero_nat) = vEBT_Leaf($false,(B2)) ).

% vebt_delete.simps(1)
tff(fact_2098_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu: nat] : vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary2),Uu) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary2) ).

% vebt_delete.simps(4)
tff(fact_2099_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X2),N) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X_1)
               => ( ! [X2: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList))
                     => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X2),X_1) )
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary2)),Deg2) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_2100_vebt__delete_Osimps_I2_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_vebt_delete(vEBT_Leaf((A2),(B2)),aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf((A2),$false) ).

% vebt_delete.simps(2)
tff(fact_2101_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList),Vd)),Xb)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_2102_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList),S)),Xb)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_2103_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V),TreeList),Vc)),Xb)
    <=> ( ( Xb = Mi )
        | ( Xb = Ma )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_2104_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi: nat,Ma: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X2),N) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = N )
           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [I2: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                   => ( ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I2)),X_12)
                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),I2) ) )
               => ( ( ( Mi = Ma )
                   => ! [X2: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList))
                       => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X2),X_1) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I2 )
                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                & ! [X2: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N) = I2 )
                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X2)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Ma) ) ) ) ) )
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),Deg2) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_2105_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi: nat,Ma: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X2),N) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,N) )
           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M) )
             => ( ! [I2: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                   => ( ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I2)),X_12)
                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),I2) ) )
               => ( ( ( Mi = Ma )
                   => ! [X2: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList))
                       => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X2),X_1) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I2 )
                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                & ! [X2: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N) = I2 )
                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X2)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Ma) ) ) ) ) )
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),Deg2) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_2106_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,Xb: nat] : vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TrLst),Smry),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TrLst),Smry) ).

% vebt_delete.simps(5)
tff(fact_2107_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,Va3),Vb2)
           => ~ ( ( Xaa = Mi2 )
                | ( Xaa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)
             => ~ ( ( Xaa = Mi2 )
                  | ( Xaa = Ma2 )
                  | $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)
               => ~ $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_2108_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,N: nat,Va: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),N)
     => ( ( N = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Mi)
         => ( ( Ma != Mi )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(TreeList)) ) ) ) ) ).

% nested_mint
tff(fact_2109_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Xb),Xaa) = Y )
     => ( ( ? [Uu2: $o,B: $o] : Xb = vEBT_Leaf((Uu2),(B))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : Xb = vEBT_Leaf((Uv),(Uw))
           => ( ? [N3: nat] : Xaa = aa_nat_nat(suc,N3)
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(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] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( Y != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                            one_one_nat,
                            $let(
                              l: nat,
                              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                  $let(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,nth_VEBT_VEBT(TreeList2,h)),l)),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Summary),h)),one_one_nat) ) ),
                                  one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_2110_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Xb),Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv: $o] : Xb = vEBT_Leaf((Uu2),(Uv))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A: $o,Uw: $o] : Xb = vEBT_Leaf((A),(Uw))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A: $o,B: $o] : Xb = vEBT_Leaf((A),(B))
             => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(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] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( Y != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                              one_one_nat,
                              $let(
                                l: nat,
                                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                    $let(
                                      minlow: option_nat,
                                      minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                      $ite(
                                        ( ( minlow != none_nat )
                                        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,nth_VEBT_VEBT(TreeList2,h)),l)),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Summary),h)),one_one_nat) ) ),
                                    one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_2111_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,Xb),Xaa) = Y )
     => ( ( ? [A: $o,B: $o] : Xb = vEBT_Leaf((A),(B))
         => ( Y != one_one_nat ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)
           => ( Y != one_one_nat ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(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] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                          $ite(
                            Xaa = Mi2,
                            zero_zero_nat,
                            $ite(
                              Xaa = Ma2,
                              zero_zero_nat,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                zero_zero_nat,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  zero_zero_nat,
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Xaa)
                                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Ma2) ),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_nat) ),
                                    zero_zero_nat ) ) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
tff(fact_2112_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2)),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi),
        one_one_nat,
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
              $let(
                maxlow: option_nat,
                maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                $ite(
                  ( ( maxlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,nth_VEBT_VEBT(TreeList,h)),l)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Summary2),h)),one_one_nat) ) ),
              one_one_nat ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_2113_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2)),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xb),
        one_one_nat,
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
              $let(
                minlow: option_nat,
                minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                $ite(
                  ( ( minlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,nth_VEBT_VEBT(TreeList,h)),l)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Summary2),h)),one_one_nat) ) ),
              one_one_nat ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_2114_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,Summary2: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2)),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(
          Xb = Mi,
          zero_zero_nat,
          $ite(
            Xb = Ma,
            zero_zero_nat,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi),
              zero_zero_nat,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xb),
                zero_zero_nat,
                $ite(
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xb)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Ma) ),
                  $let(
                    h: nat,
                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_nat) ),
                  zero_zero_nat ) ) ) ) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
tff(fact_2115_nat__induct2,axiom,
    ! [P: fun_nat_o,N: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N3: nat] :
              ( aa_nat_o(P,N3)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => aa_nat_o(P,N) ) ) ) ).

% nat_induct2
tff(fact_2116_field__less__half__sum,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_2117_field__less__half__sum,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),divide_divide_rat(plus_plus_rat(Xb,Y),numeral_numeral_rat(aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_2118_bit__concat__def,axiom,
    ! [H: nat,L: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(H),L),D) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),D))),L) ).

% bit_concat_def
tff(fact_2119_mul__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,Xb)),aa_nat_option_nat(some_nat,Y)) = aa_nat_option_nat(some_nat,Z) ) ) ).

% mul_shift
tff(fact_2120_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_2121_high__inv,axiom,
    ! [Xb: nat,N: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),Xb)),N) = Y ) ) ).

% high_inv
tff(fact_2122_low__inv,axiom,
    ! [Xb: nat,N: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),Xb)),N) = Xb ) ) ).

% low_inv
tff(fact_2123_mult__zero__left,axiom,
    ! [A2: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(zero_z3563351764282998399l_num1),A2) = zero_z3563351764282998399l_num1 ).

% mult_zero_left
tff(fact_2124_mult__zero__left,axiom,
    ! [A2: real] : aa_real_real(times_times_real(zero_zero_real),A2) = zero_zero_real ).

% mult_zero_left
tff(fact_2125_mult__zero__left,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A2) = zero_zero_rat ).

% mult_zero_left
tff(fact_2126_mult__zero__left,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A2) = zero_zero_nat ).

% mult_zero_left
tff(fact_2127_mult__zero__left,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),A2) = zero_zero_int ).

% mult_zero_left
tff(fact_2128_mult__zero__right,axiom,
    ! [A2: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A2),zero_z3563351764282998399l_num1) = zero_z3563351764282998399l_num1 ).

% mult_zero_right
tff(fact_2129_mult__zero__right,axiom,
    ! [A2: real] : aa_real_real(times_times_real(A2),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_2130_mult__zero__right,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(A2),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_2131_mult__zero__right,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),zero_zero_nat) = zero_zero_nat ).

% mult_zero_right
tff(fact_2132_mult__zero__right,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_2133_mult__eq__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(times_times_real(A2),B2) = zero_zero_real )
    <=> ( ( A2 = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_2134_mult__eq__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),B2) = zero_zero_rat )
    <=> ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_2135_mult__eq__0__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_2136_mult__eq__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) = zero_zero_int )
    <=> ( ( A2 = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_2137_mult__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( aa_real_real(times_times_real(C),A2) = aa_real_real(times_times_real(C),B2) )
    <=> ( ( C = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_2138_mult__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A2) = aa_rat_rat(times_times_rat(C),B2) )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_2139_mult__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2) )
    <=> ( ( C = zero_zero_nat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_2140_mult__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) )
    <=> ( ( C = zero_zero_int )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_2141_mult__cancel__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( aa_real_real(times_times_real(A2),C) = aa_real_real(times_times_real(B2),C) )
    <=> ( ( C = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_2142_mult__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),C) = aa_rat_rat(times_times_rat(B2),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_2143_mult__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_2144_mult__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),C) )
    <=> ( ( C = zero_zero_int )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_2145_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(V)),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(W)),Z)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_2146_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(times_times_real(numeral_numeral_real(V)),aa_real_real(times_times_real(numeral_numeral_real(W)),Z)) = aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_2147_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),Z)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_2148_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_2149_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_2150_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(M)),numera7442385471795722001l_num1(N)) = numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_2151_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(times_times_real(numeral_numeral_real(M)),numeral_numeral_real(N)) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_2152_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(numeral_numeral_rat(M)),numeral_numeral_rat(N)) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_2153_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_2154_numeral__times__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% numeral_times_numeral
tff(fact_2155_mult_Oright__neutral,axiom,
    ! [A2: uint32] : aa_uint32_uint32(times_times_uint32(A2),one_one_uint32) = A2 ).

% mult.right_neutral
tff(fact_2156_mult_Oright__neutral,axiom,
    ! [A2: real] : aa_real_real(times_times_real(A2),one_one_real) = A2 ).

% mult.right_neutral
tff(fact_2157_mult_Oright__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(A2),one_one_rat) = A2 ).

% mult.right_neutral
tff(fact_2158_mult_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),one_one_nat) = A2 ).

% mult.right_neutral
tff(fact_2159_mult_Oright__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),one_one_int) = A2 ).

% mult.right_neutral
tff(fact_2160_mult__1,axiom,
    ! [A2: uint32] : aa_uint32_uint32(times_times_uint32(one_one_uint32),A2) = A2 ).

% mult_1
tff(fact_2161_mult__1,axiom,
    ! [A2: real] : aa_real_real(times_times_real(one_one_real),A2) = A2 ).

% mult_1
tff(fact_2162_mult__1,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(one_one_rat),A2) = A2 ).

% mult_1
tff(fact_2163_mult__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A2) = A2 ).

% mult_1
tff(fact_2164_mult__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A2) = A2 ).

% mult_1
tff(fact_2165_times__divide__eq__left,axiom,
    ! [B2: real,C: real,A2: real] : aa_real_real(times_times_real(aa_real_real(divide_divide_real(B2),C)),A2) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(B2),A2)),C) ).

% times_divide_eq_left
tff(fact_2166_times__divide__eq__left,axiom,
    ! [B2: rat,C: rat,A2: rat] : aa_rat_rat(times_times_rat(divide_divide_rat(B2,C)),A2) = divide_divide_rat(aa_rat_rat(times_times_rat(B2),A2),C) ).

% times_divide_eq_left
tff(fact_2167_divide__divide__eq__left,axiom,
    ! [A2: real,B2: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(A2),B2)),C) = aa_real_real(divide_divide_real(A2),aa_real_real(times_times_real(B2),C)) ).

% divide_divide_eq_left
tff(fact_2168_divide__divide__eq__left,axiom,
    ! [A2: rat,B2: rat,C: rat] : divide_divide_rat(divide_divide_rat(A2,B2),C) = divide_divide_rat(A2,aa_rat_rat(times_times_rat(B2),C)) ).

% divide_divide_eq_left
tff(fact_2169_divide__divide__eq__right,axiom,
    ! [A2: real,B2: real,C: real] : aa_real_real(divide_divide_real(A2),aa_real_real(divide_divide_real(B2),C)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(A2),C)),B2) ).

% divide_divide_eq_right
tff(fact_2170_divide__divide__eq__right,axiom,
    ! [A2: rat,B2: rat,C: rat] : divide_divide_rat(A2,divide_divide_rat(B2,C)) = divide_divide_rat(aa_rat_rat(times_times_rat(A2),C),B2) ).

% divide_divide_eq_right
tff(fact_2171_times__divide__eq__right,axiom,
    ! [A2: real,B2: real,C: real] : aa_real_real(times_times_real(A2),aa_real_real(divide_divide_real(B2),C)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(A2),B2)),C) ).

% times_divide_eq_right
tff(fact_2172_times__divide__eq__right,axiom,
    ! [A2: rat,B2: rat,C: rat] : aa_rat_rat(times_times_rat(A2),divide_divide_rat(B2,C)) = divide_divide_rat(aa_rat_rat(times_times_rat(A2),B2),C) ).

% times_divide_eq_right
tff(fact_2173_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_2174_mult__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_2175_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
    <=> ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_2176_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K) )
    <=> ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_2177_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_2178_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_2179_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% semiring_norm(6)
tff(fact_2180_sum__squares__eq__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y),Y)) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2181_sum__squares__eq__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb),aa_rat_rat(times_times_rat(Y),Y)) = zero_zero_rat )
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2182_sum__squares__eq__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2183_mult__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( ( C = aa_real_real(times_times_real(C),B2) )
    <=> ( ( C = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_2184_mult__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( ( C = aa_rat_rat(times_times_rat(C),B2) )
    <=> ( ( C = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_2185_mult__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) )
    <=> ( ( C = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_2186_mult__cancel__left2,axiom,
    ! [C: real,A2: real] :
      ( ( aa_real_real(times_times_real(C),A2) = C )
    <=> ( ( C = zero_zero_real )
        | ( A2 = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_2187_mult__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A2) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_2188_mult__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A2) = C )
    <=> ( ( C = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_2189_mult__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( ( C = aa_real_real(times_times_real(B2),C) )
    <=> ( ( C = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_2190_mult__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( ( C = aa_rat_rat(times_times_rat(B2),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_2191_mult__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B2),C) )
    <=> ( ( C = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_2192_mult__cancel__right2,axiom,
    ! [A2: real,C: real] :
      ( ( aa_real_real(times_times_real(A2),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( A2 = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_2193_mult__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_2194_mult__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_2195_distrib__right__numeral,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,V: num] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(plus_p361126936061061375l_num1(A2,B2)),numera7442385471795722001l_num1(V)) = plus_p361126936061061375l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A2),numera7442385471795722001l_num1(V)),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B2),numera7442385471795722001l_num1(V))) ).

% distrib_right_numeral
tff(fact_2196_distrib__right__numeral,axiom,
    ! [A2: real,B2: real,V: num] : aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),numeral_numeral_real(V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A2),numeral_numeral_real(V))),aa_real_real(times_times_real(B2),numeral_numeral_real(V))) ).

% distrib_right_numeral
tff(fact_2197_distrib__right__numeral,axiom,
    ! [A2: rat,B2: rat,V: num] : aa_rat_rat(times_times_rat(plus_plus_rat(A2,B2)),numeral_numeral_rat(V)) = plus_plus_rat(aa_rat_rat(times_times_rat(A2),numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(B2),numeral_numeral_rat(V))) ).

% distrib_right_numeral
tff(fact_2198_distrib__right__numeral,axiom,
    ! [A2: nat,B2: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_num_nat(numeral_numeral_nat,V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_num_nat(numeral_numeral_nat,V))) ).

% distrib_right_numeral
tff(fact_2199_distrib__right__numeral,axiom,
    ! [A2: int,B2: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_2200_distrib__left__numeral,axiom,
    ! [V: num,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(V)),plus_p361126936061061375l_num1(B2,C)) = plus_p361126936061061375l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(V)),B2),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(V)),C)) ).

% distrib_left_numeral
tff(fact_2201_distrib__left__numeral,axiom,
    ! [V: num,B2: real,C: real] : aa_real_real(times_times_real(numeral_numeral_real(V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(numeral_numeral_real(V)),B2)),aa_real_real(times_times_real(numeral_numeral_real(V)),C)) ).

% distrib_left_numeral
tff(fact_2202_distrib__left__numeral,axiom,
    ! [V: num,B2: rat,C: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),plus_plus_rat(B2,C)) = plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),B2),aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),C)) ).

% distrib_left_numeral
tff(fact_2203_distrib__left__numeral,axiom,
    ! [V: num,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C)) ).

% distrib_left_numeral
tff(fact_2204_distrib__left__numeral,axiom,
    ! [V: num,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% distrib_left_numeral
tff(fact_2205_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A2: real,B2: real] :
      aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2)) = $ite(C = zero_zero_real,zero_zero_real,aa_real_real(divide_divide_real(A2),B2)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2206_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      divide_divide_rat(aa_rat_rat(times_times_rat(C),A2),aa_rat_rat(times_times_rat(C),B2)) = $ite(C = zero_zero_rat,zero_zero_rat,divide_divide_rat(A2,B2)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2207_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2)) = aa_real_real(divide_divide_real(A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2208_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A2),aa_rat_rat(times_times_rat(C),B2)) = divide_divide_rat(A2,B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2209_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(B2),C)) = aa_real_real(divide_divide_real(A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2210_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(C),A2),aa_rat_rat(times_times_rat(B2),C)) = divide_divide_rat(A2,B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2211_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),C)) = aa_real_real(divide_divide_real(A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2212_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A2),C),aa_rat_rat(times_times_rat(B2),C)) = divide_divide_rat(A2,B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2213_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(C),B2)) = aa_real_real(divide_divide_real(A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2214_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A2),C),aa_rat_rat(times_times_rat(C),B2)) = divide_divide_rat(A2,B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2215_div__mult__mult1,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) ) ) ).

% div_mult_mult1
tff(fact_2216_div__mult__mult1,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ) ) ).

% div_mult_mult1
tff(fact_2217_div__mult__mult2,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) ) ) ).

% div_mult_mult2
tff(fact_2218_div__mult__mult2,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ) ) ).

% div_mult_mult2
tff(fact_2219_div__mult__mult1__if,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ).

% div_mult_mult1_if
tff(fact_2220_div__mult__mult1__if,axiom,
    ! [C: int,A2: int,B2: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) = $ite(C = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ).

% div_mult_mult1_if
tff(fact_2221_nonzero__mult__div__cancel__left,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A2),B2)),A2) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2222_nonzero__mult__div__cancel__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A2),B2),A2) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2223_nonzero__mult__div__cancel__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),A2) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2224_nonzero__mult__div__cancel__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),A2) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2225_nonzero__mult__div__cancel__right,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A2),B2)),B2) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2226_nonzero__mult__div__cancel__right,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( divide_divide_rat(aa_rat_rat(times_times_rat(A2),B2),B2) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2227_nonzero__mult__div__cancel__right,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),B2) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2228_nonzero__mult__div__cancel__right,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),B2) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2229_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(V)),minus_4019991460397169231l_num1(B2,C)) = minus_4019991460397169231l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(V)),B2),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_2230_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: real,C: real] : aa_real_real(times_times_real(numeral_numeral_real(V)),aa_real_real(minus_minus_real(B2),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(numeral_numeral_real(V)),B2)),aa_real_real(times_times_real(numeral_numeral_real(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_2231_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: rat,C: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),minus_minus_rat(B2,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),B2),aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_2232_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_2233_left__diff__distrib__numeral,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,V: num] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(minus_4019991460397169231l_num1(A2,B2)),numera7442385471795722001l_num1(V)) = minus_4019991460397169231l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A2),numera7442385471795722001l_num1(V)),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B2),numera7442385471795722001l_num1(V))) ).

% left_diff_distrib_numeral
tff(fact_2234_left__diff__distrib__numeral,axiom,
    ! [A2: real,B2: real,V: num] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(A2),B2)),numeral_numeral_real(V)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A2),numeral_numeral_real(V))),aa_real_real(times_times_real(B2),numeral_numeral_real(V))) ).

% left_diff_distrib_numeral
tff(fact_2235_left__diff__distrib__numeral,axiom,
    ! [A2: rat,B2: rat,V: num] : aa_rat_rat(times_times_rat(minus_minus_rat(A2,B2)),numeral_numeral_rat(V)) = minus_minus_rat(aa_rat_rat(times_times_rat(A2),numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(B2),numeral_numeral_rat(V))) ).

% left_diff_distrib_numeral
tff(fact_2236_left__diff__distrib__numeral,axiom,
    ! [A2: int,B2: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_num_int(numeral_numeral_int,V))) ).

% left_diff_distrib_numeral
tff(fact_2237_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_2238_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( N = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_2239_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_2240_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% mult_less_cancel2
tff(fact_2241_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% nat_0_less_mult_iff
tff(fact_2242_mult__Suc__right,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc_right
tff(fact_2243_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = $ite(K = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) ).

% nat_mult_div_cancel_disj
tff(fact_2244_semiring__norm_I2_J,axiom,
    aa_num_num(aa_num_fun_num_num(plus_plus_num,one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_2245_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% zle_add1_eq_le
tff(fact_2246_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( ( aa_real_real(divide_divide_real(B2),numeral_numeral_real(W)) = A2 )
    <=> $ite(numeral_numeral_real(W) != zero_zero_real,B2 = aa_real_real(times_times_real(A2),numeral_numeral_real(W)),A2 = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2247_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( divide_divide_rat(B2,numeral_numeral_rat(W)) = A2 )
    <=> $ite(numeral_numeral_rat(W) != zero_zero_rat,B2 = aa_rat_rat(times_times_rat(A2),numeral_numeral_rat(W)),A2 = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2248_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( A2 = aa_real_real(divide_divide_real(B2),numeral_numeral_real(W)) )
    <=> $ite(numeral_numeral_real(W) != zero_zero_real,aa_real_real(times_times_real(A2),numeral_numeral_real(W)) = B2,A2 = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2249_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( A2 = divide_divide_rat(B2,numeral_numeral_rat(W)) )
    <=> $ite(numeral_numeral_rat(W) != zero_zero_rat,aa_rat_rat(times_times_rat(A2),numeral_numeral_rat(W)) = B2,A2 = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2250_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B2),numeral_numeral_real(W))),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(times_times_real(A2),numeral_numeral_real(W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2251_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B2,numeral_numeral_rat(W))),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),numeral_numeral_rat(W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2252_le__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(divide_divide_real(B2),numeral_numeral_real(W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),numeral_numeral_real(W))),B2) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2253_le__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),divide_divide_rat(B2,numeral_numeral_rat(W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),numeral_numeral_rat(W))),B2) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2254_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B2),numeral_numeral_real(W))),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(times_times_real(A2),numeral_numeral_real(W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_2255_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B2,numeral_numeral_rat(W))),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),numeral_numeral_rat(W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_2256_less__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(divide_divide_real(B2),numeral_numeral_real(W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),numeral_numeral_real(W))),B2) ) ).

% less_divide_eq_numeral1(1)
tff(fact_2257_less__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),divide_divide_rat(B2,numeral_numeral_rat(W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),numeral_numeral_rat(W))),B2) ) ).

% less_divide_eq_numeral1(1)
tff(fact_2258_div__mult__self1,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ) ) ).

% div_mult_self1
tff(fact_2259_div__mult__self1,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% div_mult_self1
tff(fact_2260_div__mult__self2,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ) ) ).

% div_mult_self2
tff(fact_2261_div__mult__self2,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% div_mult_self2
tff(fact_2262_div__mult__self3,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2)),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ) ) ).

% div_mult_self3
tff(fact_2263_div__mult__self3,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)),A2)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% div_mult_self3
tff(fact_2264_div__mult__self4,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ) ) ).

% div_mult_self4
tff(fact_2265_div__mult__self4,axiom,
    ! [B2: int,C: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)),A2)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% div_mult_self4
tff(fact_2266_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A2),aa_real_real(times_times_real(A2),B2)) = aa_real_real(divide_divide_real(one_one_real),B2) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2267_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( divide_divide_rat(A2,aa_rat_rat(times_times_rat(A2),B2)) = divide_divide_rat(one_one_rat,B2) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2268_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(divide_divide_real(B2),aa_real_real(times_times_real(A2),B2)) = aa_real_real(divide_divide_real(one_one_real),A2) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2269_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( divide_divide_rat(B2,aa_rat_rat(times_times_rat(A2),B2)) = divide_divide_rat(one_one_rat,A2) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2270_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),N) ) ) ).

% one_le_mult_iff
tff(fact_2271_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_2272_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% mult_le_cancel2
tff(fact_2273_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),N) = M ) ) ).

% div_mult_self_is_m
tff(fact_2274_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),N) = M ) ) ).

% div_mult_self1_is_m
tff(fact_2275_Suc__numeral,axiom,
    ! [N: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% Suc_numeral
tff(fact_2276_power__add__numeral,axiom,
    ! [A2: complex,M: num,N: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_2277_power__add__numeral,axiom,
    ! [A2: code_integer,M: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_2278_power__add__numeral,axiom,
    ! [A2: real,M: num,N: num] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_2279_power__add__numeral,axiom,
    ! [A2: rat,M: num,N: num] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_2280_power__add__numeral,axiom,
    ! [A2: nat,M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_2281_power__add__numeral,axiom,
    ! [A2: int,M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,N))) = aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% power_add_numeral
tff(fact_2282_power__add__numeral2,axiom,
    ! [A2: complex,M: num,N: num,B2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,M))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,N))),B2)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B2) ).

% power_add_numeral2
tff(fact_2283_power__add__numeral2,axiom,
    ! [A2: code_integer,M: num,N: num,B2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,M))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,N))),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B2) ).

% power_add_numeral2
tff(fact_2284_power__add__numeral2,axiom,
    ! [A2: real,M: num,N: num,B2: real] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,M))),aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,N))),B2)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B2) ).

% power_add_numeral2
tff(fact_2285_power__add__numeral2,axiom,
    ! [A2: rat,M: num,N: num,B2: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,M))),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,N))),B2)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B2) ).

% power_add_numeral2
tff(fact_2286_power__add__numeral2,axiom,
    ! [A2: nat,M: num,N: num,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,N))),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B2) ).

% power_add_numeral2
tff(fact_2287_power__add__numeral2,axiom,
    ! [A2: int,M: num,N: num,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,N))),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)))),B2) ).

% power_add_numeral2
tff(fact_2288_mult__not__zero,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A2),B2) != zero_z3563351764282998399l_num1 )
     => ( ( A2 != zero_z3563351764282998399l_num1 )
        & ( B2 != zero_z3563351764282998399l_num1 ) ) ) ).

% mult_not_zero
tff(fact_2289_mult__not__zero,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(times_times_real(A2),B2) != zero_zero_real )
     => ( ( A2 != zero_zero_real )
        & ( B2 != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_2290_mult__not__zero,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),B2) != zero_zero_rat )
     => ( ( A2 != zero_zero_rat )
        & ( B2 != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_2291_mult__not__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) != zero_zero_nat )
     => ( ( A2 != zero_zero_nat )
        & ( B2 != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_2292_mult__not__zero,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) != zero_zero_int )
     => ( ( A2 != zero_zero_int )
        & ( B2 != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_2293_divisors__zero,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(times_times_real(A2),B2) = zero_zero_real )
     => ( ( A2 = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_2294_divisors__zero,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),B2) = zero_zero_rat )
     => ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_2295_divisors__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = zero_zero_nat )
     => ( ( A2 = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_2296_divisors__zero,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) = zero_zero_int )
     => ( ( A2 = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_2297_no__zero__divisors,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( aa_real_real(times_times_real(A2),B2) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_2298_no__zero__divisors,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( aa_rat_rat(times_times_rat(A2),B2) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_2299_no__zero__divisors,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( B2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_2300_no__zero__divisors,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( B2 != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_2301_mult__left__cancel,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(C),A2) = aa_real_real(times_times_real(C),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_2302_mult__left__cancel,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C),A2) = aa_rat_rat(times_times_rat(C),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_2303_mult__left__cancel,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_2304_mult__left__cancel,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_2305_mult__right__cancel,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(A2),C) = aa_real_real(times_times_real(B2),C) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_2306_mult__right__cancel,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A2),C) = aa_rat_rat(times_times_rat(B2),C) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_2307_mult__right__cancel,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_2308_mult__right__cancel,axiom,
    ! [C: int,A2: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),C) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_2309_add__One__commute,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N) = aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one) ).

% add_One_commute
tff(fact_2310_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: real,B2: real,C: real] : aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2311_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: rat,B2: rat,C: rat] : aa_rat_rat(times_times_rat(plus_plus_rat(A2,B2)),C) = plus_plus_rat(aa_rat_rat(times_times_rat(A2),C),aa_rat_rat(times_times_rat(B2),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2312_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2313_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: real,B2: real,C: real] : aa_real_real(times_times_real(A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A2),B2)),aa_real_real(times_times_real(A2),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2314_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: rat,B2: rat,C: rat] : aa_rat_rat(times_times_rat(A2),plus_plus_rat(B2,C)) = plus_plus_rat(aa_rat_rat(times_times_rat(A2),B2),aa_rat_rat(times_times_rat(A2),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2315_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2316_comm__semiring__class_Odistrib,axiom,
    ! [A2: real,B2: real,C: real] : aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),C)) ).

% comm_semiring_class.distrib
tff(fact_2317_comm__semiring__class_Odistrib,axiom,
    ! [A2: rat,B2: rat,C: rat] : aa_rat_rat(times_times_rat(plus_plus_rat(A2,B2)),C) = plus_plus_rat(aa_rat_rat(times_times_rat(A2),C),aa_rat_rat(times_times_rat(B2),C)) ).

% comm_semiring_class.distrib
tff(fact_2318_comm__semiring__class_Odistrib,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) ).

% comm_semiring_class.distrib
tff(fact_2319_comm__semiring__class_Odistrib,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ).

% comm_semiring_class.distrib
tff(fact_2320_distrib__left,axiom,
    ! [A2: real,B2: real,C: real] : aa_real_real(times_times_real(A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A2),B2)),aa_real_real(times_times_real(A2),C)) ).

% distrib_left
tff(fact_2321_distrib__left,axiom,
    ! [A2: rat,B2: rat,C: rat] : aa_rat_rat(times_times_rat(A2),plus_plus_rat(B2,C)) = plus_plus_rat(aa_rat_rat(times_times_rat(A2),B2),aa_rat_rat(times_times_rat(A2),C)) ).

% distrib_left
tff(fact_2322_distrib__left,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)) ).

% distrib_left
tff(fact_2323_distrib__left,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)) ).

% distrib_left
tff(fact_2324_distrib__right,axiom,
    ! [A2: real,B2: real,C: real] : aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),C)) ).

% distrib_right
tff(fact_2325_distrib__right,axiom,
    ! [A2: rat,B2: rat,C: rat] : aa_rat_rat(times_times_rat(plus_plus_rat(A2,B2)),C) = plus_plus_rat(aa_rat_rat(times_times_rat(A2),C),aa_rat_rat(times_times_rat(B2),C)) ).

% distrib_right
tff(fact_2326_distrib__right,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) ).

% distrib_right
tff(fact_2327_distrib__right,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ).

% distrib_right
tff(fact_2328_combine__common__factor,axiom,
    ! [A2: real,E2: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A2),E2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(B2),E2)),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),E2)),C) ).

% combine_common_factor
tff(fact_2329_combine__common__factor,axiom,
    ! [A2: rat,E2: rat,B2: rat,C: rat] : plus_plus_rat(aa_rat_rat(times_times_rat(A2),E2),plus_plus_rat(aa_rat_rat(times_times_rat(B2),E2),C)) = plus_plus_rat(aa_rat_rat(times_times_rat(plus_plus_rat(A2,B2)),E2),C) ).

% combine_common_factor
tff(fact_2330_combine__common__factor,axiom,
    ! [A2: nat,E2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),E2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),E2)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),E2)),C) ).

% combine_common_factor
tff(fact_2331_combine__common__factor,axiom,
    ! [A2: int,E2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),E2)),C) ).

% combine_common_factor
tff(fact_2332_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: uint32] : aa_uint32_uint32(times_times_uint32(one_one_uint32),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_2333_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: real] : aa_real_real(times_times_real(one_one_real),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_2334_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(one_one_rat),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_2335_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_2336_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_2337_mult_Ocomm__neutral,axiom,
    ! [A2: uint32] : aa_uint32_uint32(times_times_uint32(A2),one_one_uint32) = A2 ).

% mult.comm_neutral
tff(fact_2338_mult_Ocomm__neutral,axiom,
    ! [A2: real] : aa_real_real(times_times_real(A2),one_one_real) = A2 ).

% mult.comm_neutral
tff(fact_2339_mult_Ocomm__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(A2),one_one_rat) = A2 ).

% mult.comm_neutral
tff(fact_2340_mult_Ocomm__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),one_one_nat) = A2 ).

% mult.comm_neutral
tff(fact_2341_mult_Ocomm__neutral,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),one_one_int) = A2 ).

% mult.comm_neutral
tff(fact_2342_left__diff__distrib,axiom,
    ! [A2: real,B2: real,C: real] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(A2),B2)),C) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),C)) ).

% left_diff_distrib
tff(fact_2343_left__diff__distrib,axiom,
    ! [A2: rat,B2: rat,C: rat] : aa_rat_rat(times_times_rat(minus_minus_rat(A2,B2)),C) = minus_minus_rat(aa_rat_rat(times_times_rat(A2),C),aa_rat_rat(times_times_rat(B2),C)) ).

% left_diff_distrib
tff(fact_2344_left__diff__distrib,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ).

% left_diff_distrib
tff(fact_2345_right__diff__distrib,axiom,
    ! [A2: real,B2: real,C: real] : aa_real_real(times_times_real(A2),aa_real_real(minus_minus_real(B2),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A2),B2)),aa_real_real(times_times_real(A2),C)) ).

% right_diff_distrib
tff(fact_2346_right__diff__distrib,axiom,
    ! [A2: rat,B2: rat,C: rat] : aa_rat_rat(times_times_rat(A2),minus_minus_rat(B2,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(A2),B2),aa_rat_rat(times_times_rat(A2),C)) ).

% right_diff_distrib
tff(fact_2347_right__diff__distrib,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)) ).

% right_diff_distrib
tff(fact_2348_left__diff__distrib_H,axiom,
    ! [B2: real,C: real,A2: real] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(B2),C)),A2) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(B2),A2)),aa_real_real(times_times_real(C),A2)) ).

% left_diff_distrib'
tff(fact_2349_left__diff__distrib_H,axiom,
    ! [B2: rat,C: rat,A2: rat] : aa_rat_rat(times_times_rat(minus_minus_rat(B2,C)),A2) = minus_minus_rat(aa_rat_rat(times_times_rat(B2),A2),aa_rat_rat(times_times_rat(C),A2)) ).

% left_diff_distrib'
tff(fact_2350_left__diff__distrib_H,axiom,
    ! [B2: nat,C: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),C)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)) ).

% left_diff_distrib'
tff(fact_2351_left__diff__distrib_H,axiom,
    ! [B2: int,C: int,A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)),A2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)) ).

% left_diff_distrib'
tff(fact_2352_right__diff__distrib_H,axiom,
    ! [A2: real,B2: real,C: real] : aa_real_real(times_times_real(A2),aa_real_real(minus_minus_real(B2),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A2),B2)),aa_real_real(times_times_real(A2),C)) ).

% right_diff_distrib'
tff(fact_2353_right__diff__distrib_H,axiom,
    ! [A2: rat,B2: rat,C: rat] : aa_rat_rat(times_times_rat(A2),minus_minus_rat(B2,C)) = minus_minus_rat(aa_rat_rat(times_times_rat(A2),B2),aa_rat_rat(times_times_rat(A2),C)) ).

% right_diff_distrib'
tff(fact_2354_right__diff__distrib_H,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)) ).

% right_diff_distrib'
tff(fact_2355_right__diff__distrib_H,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)) ).

% right_diff_distrib'
tff(fact_2356_times__divide__times__eq,axiom,
    ! [Xb: real,Y: real,Z: real,W: real] : aa_real_real(times_times_real(aa_real_real(divide_divide_real(Xb),Y)),aa_real_real(divide_divide_real(Z),W)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(Y),W)) ).

% times_divide_times_eq
tff(fact_2357_times__divide__times__eq,axiom,
    ! [Xb: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(times_times_rat(divide_divide_rat(Xb,Y)),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(times_times_rat(Xb),Z),aa_rat_rat(times_times_rat(Y),W)) ).

% times_divide_times_eq
tff(fact_2358_divide__divide__times__eq,axiom,
    ! [Xb: real,Y: real,Z: real,W: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(Xb),Y)),aa_real_real(divide_divide_real(Z),W)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(Xb),W)),aa_real_real(times_times_real(Y),Z)) ).

% divide_divide_times_eq
tff(fact_2359_divide__divide__times__eq,axiom,
    ! [Xb: rat,Y: rat,Z: rat,W: rat] : divide_divide_rat(divide_divide_rat(Xb,Y),divide_divide_rat(Z,W)) = divide_divide_rat(aa_rat_rat(times_times_rat(Xb),W),aa_rat_rat(times_times_rat(Y),Z)) ).

% divide_divide_times_eq
tff(fact_2360_divide__divide__eq__left_H,axiom,
    ! [A2: real,B2: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(A2),B2)),C) = aa_real_real(divide_divide_real(A2),aa_real_real(times_times_real(C),B2)) ).

% divide_divide_eq_left'
tff(fact_2361_divide__divide__eq__left_H,axiom,
    ! [A2: rat,B2: rat,C: rat] : divide_divide_rat(divide_divide_rat(A2,B2),C) = divide_divide_rat(A2,aa_rat_rat(times_times_rat(C),B2)) ).

% divide_divide_eq_left'
tff(fact_2362_power__commutes,axiom,
    ! [A2: complex,N: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)),A2) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)) ).

% power_commutes
tff(fact_2363_power__commutes,axiom,
    ! [A2: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ).

% power_commutes
tff(fact_2364_power__commutes,axiom,
    ! [A2: real,N: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),A2) = aa_real_real(times_times_real(A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ).

% power_commutes
tff(fact_2365_power__commutes,axiom,
    ! [A2: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),N)),A2) = aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(A2),N)) ).

% power_commutes
tff(fact_2366_power__commutes,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ).

% power_commutes
tff(fact_2367_power__commutes,axiom,
    ! [A2: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A2),N)),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(power_power_int(A2),N)) ).

% power_commutes
tff(fact_2368_power__mult__distrib,axiom,
    ! [A2: complex,B2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2)),N) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,B2),N)) ).

% power_mult_distrib
tff(fact_2369_power__mult__distrib,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N)) ).

% power_mult_distrib
tff(fact_2370_power__mult__distrib,axiom,
    ! [A2: real,B2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(times_times_real(A2),B2)),N) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)) ).

% power_mult_distrib
tff(fact_2371_power__mult__distrib,axiom,
    ! [A2: rat,B2: rat,N: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(times_times_rat(A2),B2)),N) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B2),N)) ).

% power_mult_distrib
tff(fact_2372_power__mult__distrib,axiom,
    ! [A2: nat,B2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)) ).

% power_mult_distrib
tff(fact_2373_power__mult__distrib,axiom,
    ! [A2: int,B2: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(B2),N)) ).

% power_mult_distrib
tff(fact_2374_power__commuting__commutes,axiom,
    ! [Xb: complex,Y: complex,N: nat] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),Xb) )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),N)),Y) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),N)) ) ) ).

% power_commuting_commutes
tff(fact_2375_power__commuting__commutes,axiom,
    ! [Xb: code_integer,Y: code_integer,N: nat] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Xb) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),N)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),N)) ) ) ).

% power_commuting_commutes
tff(fact_2376_power__commuting__commutes,axiom,
    ! [Xb: real,Y: real,N: nat] :
      ( ( aa_real_real(times_times_real(Xb),Y) = aa_real_real(times_times_real(Y),Xb) )
     => ( aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)),Y) = aa_real_real(times_times_real(Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)) ) ) ).

% power_commuting_commutes
tff(fact_2377_power__commuting__commutes,axiom,
    ! [Xb: rat,Y: rat,N: nat] :
      ( ( aa_rat_rat(times_times_rat(Xb),Y) = aa_rat_rat(times_times_rat(Y),Xb) )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),N)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(power_power_rat(Xb),N)) ) ) ).

% power_commuting_commutes
tff(fact_2378_power__commuting__commutes,axiom,
    ! [Xb: nat,Y: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),Xb) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),N)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),N)) ) ) ).

% power_commuting_commutes
tff(fact_2379_power__commuting__commutes,axiom,
    ! [Xb: int,Y: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb) )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),N)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(power_power_int(Xb),N)) ) ) ).

% power_commuting_commutes
tff(fact_2380_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N) )
    <=> ( M = N ) ) ).

% Suc_mult_cancel1
tff(fact_2381_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
    <=> ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_2382_mult__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),N) = zero_zero_nat ).

% mult_0
tff(fact_2383_power__mult,axiom,
    ! [A2: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),N) ).

% power_mult
tff(fact_2384_power__mult,axiom,
    ! [A2: int,M: nat,N: nat] : aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A2),M)),N) ).

% power_mult
tff(fact_2385_power__mult,axiom,
    ! [A2: real,M: nat,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),N) ).

% power_mult
tff(fact_2386_power__mult,axiom,
    ! [A2: complex,M: nat,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),M)),N) ).

% power_mult
tff(fact_2387_power__mult,axiom,
    ! [A2: code_integer,M: nat,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),N) ).

% power_mult
tff(fact_2388_plus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),L) = L ).

% plus_int_code(2)
tff(fact_2389_plus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,K),zero_zero_int) = K ).

% plus_int_code(1)
tff(fact_2390_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)) ).

% add_mult_distrib
tff(fact_2391_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) ).

% add_mult_distrib2
tff(fact_2392_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_2393_le__cube,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).

% le_cube
tff(fact_2394_le__square,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)) ).

% le_square
tff(fact_2395_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_2396_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ).

% mult_le_mono1
tff(fact_2397_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ).

% mult_le_mono2
tff(fact_2398_nat__mult__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),N) = N ).

% nat_mult_1
tff(fact_2399_nat__mult__1__right,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),one_one_nat) = N ).

% nat_mult_1_right
tff(fact_2400_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)) ).

% diff_mult_distrib
tff(fact_2401_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) ).

% diff_mult_distrib2
tff(fact_2402_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),Q2) ).

% div_mult2_eq
tff(fact_2403_iadd__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(M,N) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        & ( N = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_2404_lambda__zero,axiom,
    aTP_Lamm_is = times_7065122842183080059l_num1(zero_z3563351764282998399l_num1) ).

% lambda_zero
tff(fact_2405_lambda__zero,axiom,
    aTP_Lamm_it = times_times_real(zero_zero_real) ).

% lambda_zero
tff(fact_2406_lambda__zero,axiom,
    aTP_Lamm_iu = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_2407_lambda__zero,axiom,
    aTP_Lamm_iv = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_2408_lambda__zero,axiom,
    aTP_Lamm_iw = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_2409_lambda__one,axiom,
    aTP_Lamm_ix = times_times_uint32(one_one_uint32) ).

% lambda_one
tff(fact_2410_lambda__one,axiom,
    aTP_Lamm_iy = times_times_real(one_one_real) ).

% lambda_one
tff(fact_2411_lambda__one,axiom,
    aTP_Lamm_iz = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_2412_lambda__one,axiom,
    aTP_Lamm_eb = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_2413_lambda__one,axiom,
    aTP_Lamm_ja = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_2414_mult__mono,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),D)) ) ) ) ) ).

% mult_mono
tff(fact_2415_mult__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),D)) ) ) ) ) ).

% mult_mono
tff(fact_2416_mult__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D)) ) ) ) ) ).

% mult_mono
tff(fact_2417_mult__mono,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_mono
tff(fact_2418_mult__mono_H,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2419_mult__mono_H,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2420_mult__mono_H,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2421_mult__mono_H,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2422_zero__le__square,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A2),A2)) ).

% zero_le_square
tff(fact_2423_zero__le__square,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),A2)) ).

% zero_le_square
tff(fact_2424_zero__le__square,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),A2)) ).

% zero_le_square
tff(fact_2425_split__mult__pos__le,axiom,
    ! [A2: real,B2: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A2),B2)) ) ).

% split_mult_pos_le
tff(fact_2426_split__mult__pos__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)) ) ).

% split_mult_pos_le
tff(fact_2427_split__mult__pos__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) ) ).

% split_mult_pos_le
tff(fact_2428_mult__left__mono__neg,axiom,
    ! [B2: real,A2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2)) ) ) ).

% mult_left_mono_neg
tff(fact_2429_mult__left__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A2)),aa_rat_rat(times_times_rat(C),B2)) ) ) ).

% mult_left_mono_neg
tff(fact_2430_mult__left__mono__neg,axiom,
    ! [B2: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) ) ) ).

% mult_left_mono_neg
tff(fact_2431_mult__nonpos__nonpos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A2),B2)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2432_mult__nonpos__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2433_mult__nonpos__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2434_mult__left__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2)) ) ) ).

% mult_left_mono
tff(fact_2435_mult__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A2)),aa_rat_rat(times_times_rat(C),B2)) ) ) ).

% mult_left_mono
tff(fact_2436_mult__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2)) ) ) ).

% mult_left_mono
tff(fact_2437_mult__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) ) ) ).

% mult_left_mono
tff(fact_2438_mult__right__mono__neg,axiom,
    ! [B2: real,A2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2439_mult__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2440_mult__right__mono__neg,axiom,
    ! [B2: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2441_mult__right__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),C)) ) ) ).

% mult_right_mono
tff(fact_2442_mult__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),C)) ) ) ).

% mult_right_mono
tff(fact_2443_mult__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) ) ) ).

% mult_right_mono
tff(fact_2444_mult__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ) ) ).

% mult_right_mono
tff(fact_2445_mult__le__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),B2)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) ) ) ).

% mult_le_0_iff
tff(fact_2446_mult__le__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) ) ) ).

% mult_le_0_iff
tff(fact_2447_mult__le__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) ) ) ) ).

% mult_le_0_iff
tff(fact_2448_split__mult__neg__le,axiom,
    ! [A2: real,B2: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),B2)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_2449_split__mult__neg__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_2450_split__mult__neg__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_2451_split__mult__neg__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_2452_mult__nonneg__nonneg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A2),B2)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2453_mult__nonneg__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2454_mult__nonneg__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2455_mult__nonneg__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2456_mult__nonneg__nonpos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),B2)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_2457_mult__nonneg__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_2458_mult__nonneg__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_2459_mult__nonneg__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_2460_mult__nonpos__nonneg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),B2)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_2461_mult__nonpos__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_2462_mult__nonpos__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_2463_mult__nonpos__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_2464_mult__nonneg__nonpos2,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(B2),A2)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2465_mult__nonneg__nonpos2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(B2),A2)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2466_mult__nonneg__nonpos2,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2467_mult__nonneg__nonpos2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2468_zero__le__mult__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A2),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_2469_zero__le__mult__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_2470_zero__le__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_2471_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2472_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A2)),aa_rat_rat(times_times_rat(C),B2)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2473_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2474_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2475_mult__neg__neg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A2),B2)) ) ) ).

% mult_neg_neg
tff(fact_2476_mult__neg__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)) ) ) ).

% mult_neg_neg
tff(fact_2477_mult__neg__neg,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) ) ) ).

% mult_neg_neg
tff(fact_2478_not__square__less__zero,axiom,
    ! [A2: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),A2)),zero_zero_real) ).

% not_square_less_zero
tff(fact_2479_not__square__less__zero,axiom,
    ! [A2: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),A2)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_2480_not__square__less__zero,axiom,
    ! [A2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),A2)),zero_zero_int) ).

% not_square_less_zero
tff(fact_2481_mult__less__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),B2)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ) ) ).

% mult_less_0_iff
tff(fact_2482_mult__less__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ) ) ).

% mult_less_0_iff
tff(fact_2483_mult__less__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ) ) ).

% mult_less_0_iff
tff(fact_2484_mult__neg__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),B2)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_2485_mult__neg__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_2486_mult__neg__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_2487_mult__neg__pos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_2488_mult__pos__neg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),B2)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_2489_mult__pos__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_2490_mult__pos__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_2491_mult__pos__neg,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_2492_mult__pos__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A2),B2)) ) ) ).

% mult_pos_pos
tff(fact_2493_mult__pos__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)) ) ) ).

% mult_pos_pos
tff(fact_2494_mult__pos__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)) ) ) ).

% mult_pos_pos
tff(fact_2495_mult__pos__pos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) ) ) ).

% mult_pos_pos
tff(fact_2496_mult__pos__neg2,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(B2),A2)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_2497_mult__pos__neg2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(B2),A2)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_2498_mult__pos__neg2,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_2499_mult__pos__neg2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_2500_zero__less__mult__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A2),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_2501_zero__less__mult__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_2502_zero__less__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_2503_zero__less__mult__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A2),B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ) ).

% zero_less_mult_pos
tff(fact_2504_zero__less__mult__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ) ).

% zero_less_mult_pos
tff(fact_2505_zero__less__mult__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2) ) ) ).

% zero_less_mult_pos
tff(fact_2506_zero__less__mult__pos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ) ).

% zero_less_mult_pos
tff(fact_2507_zero__less__mult__pos2,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(B2),A2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ) ).

% zero_less_mult_pos2
tff(fact_2508_zero__less__mult__pos2,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B2),A2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ) ).

% zero_less_mult_pos2
tff(fact_2509_zero__less__mult__pos2,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2) ) ) ).

% zero_less_mult_pos2
tff(fact_2510_zero__less__mult__pos2,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ) ).

% zero_less_mult_pos2
tff(fact_2511_mult__less__cancel__left__neg,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2512_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A2)),aa_rat_rat(times_times_rat(C),B2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2513_mult__less__cancel__left__neg,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2514_mult__less__cancel__left__pos,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2515_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A2)),aa_rat_rat(times_times_rat(C),B2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2516_mult__less__cancel__left__pos,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2517_mult__strict__left__mono__neg,axiom,
    ! [B2: real,A2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2518_mult__strict__left__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A2)),aa_rat_rat(times_times_rat(C),B2)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2519_mult__strict__left__mono__neg,axiom,
    ! [B2: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2520_mult__strict__left__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2)) ) ) ).

% mult_strict_left_mono
tff(fact_2521_mult__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A2)),aa_rat_rat(times_times_rat(C),B2)) ) ) ).

% mult_strict_left_mono
tff(fact_2522_mult__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2)) ) ) ).

% mult_strict_left_mono
tff(fact_2523_mult__strict__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) ) ) ).

% mult_strict_left_mono
tff(fact_2524_mult__less__cancel__left__disj,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2525_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A2)),aa_rat_rat(times_times_rat(C),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2526_mult__less__cancel__left__disj,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2527_mult__strict__right__mono__neg,axiom,
    ! [B2: real,A2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2528_mult__strict__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2529_mult__strict__right__mono__neg,axiom,
    ! [B2: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2530_mult__strict__right__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2531_mult__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2532_mult__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2533_mult__strict__right__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2534_mult__less__cancel__right__disj,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2535_mult__less__cancel__right__disj,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2536_mult__less__cancel__right__disj,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2537_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2538_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A2)),aa_rat_rat(times_times_rat(C),B2)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2539_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2540_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2541_less__1__mult,axiom,
    ! [M: real,N: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),N)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(times_times_real(M),N)) ) ) ).

% less_1_mult
tff(fact_2542_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),N)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(M),N)) ) ) ).

% less_1_mult
tff(fact_2543_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ) ) ).

% less_1_mult
tff(fact_2544_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M),N)) ) ) ).

% less_1_mult
tff(fact_2545_mult__numeral__1,axiom,
    ! [A2: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(one)),A2) = A2 ).

% mult_numeral_1
tff(fact_2546_mult__numeral__1,axiom,
    ! [A2: real] : aa_real_real(times_times_real(numeral_numeral_real(one)),A2) = A2 ).

% mult_numeral_1
tff(fact_2547_mult__numeral__1,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(one)),A2) = A2 ).

% mult_numeral_1
tff(fact_2548_mult__numeral__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,one)),A2) = A2 ).

% mult_numeral_1
tff(fact_2549_mult__numeral__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,one)),A2) = A2 ).

% mult_numeral_1
tff(fact_2550_mult__numeral__1__right,axiom,
    ! [A2: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A2),numera7442385471795722001l_num1(one)) = A2 ).

% mult_numeral_1_right
tff(fact_2551_mult__numeral__1__right,axiom,
    ! [A2: real] : aa_real_real(times_times_real(A2),numeral_numeral_real(one)) = A2 ).

% mult_numeral_1_right
tff(fact_2552_mult__numeral__1__right,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(A2),numeral_numeral_rat(one)) = A2 ).

% mult_numeral_1_right
tff(fact_2553_mult__numeral__1__right,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_num_nat(numeral_numeral_nat,one)) = A2 ).

% mult_numeral_1_right
tff(fact_2554_mult__numeral__1__right,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_num_int(numeral_numeral_int,one)) = A2 ).

% mult_numeral_1_right
tff(fact_2555_frac__eq__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(divide_divide_real(Xb),Y) = aa_real_real(divide_divide_real(W),Z) )
        <=> ( aa_real_real(times_times_real(Xb),Z) = aa_real_real(times_times_real(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2556_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( divide_divide_rat(Xb,Y) = divide_divide_rat(W,Z) )
        <=> ( aa_rat_rat(times_times_rat(Xb),Z) = aa_rat_rat(times_times_rat(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2557_divide__eq__eq,axiom,
    ! [B2: real,C: real,A2: real] :
      ( ( aa_real_real(divide_divide_real(B2),C) = A2 )
    <=> $ite(C != zero_zero_real,B2 = aa_real_real(times_times_real(A2),C),A2 = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_2558_divide__eq__eq,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( ( divide_divide_rat(B2,C) = A2 )
    <=> $ite(C != zero_zero_rat,B2 = aa_rat_rat(times_times_rat(A2),C),A2 = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_2559_eq__divide__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( A2 = aa_real_real(divide_divide_real(B2),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(A2),C) = B2,A2 = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_2560_eq__divide__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( A2 = divide_divide_rat(B2,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A2),C) = B2,A2 = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_2561_divide__eq__imp,axiom,
    ! [C: real,B2: real,A2: real] :
      ( ( C != zero_zero_real )
     => ( ( B2 = aa_real_real(times_times_real(A2),C) )
       => ( aa_real_real(divide_divide_real(B2),C) = A2 ) ) ) ).

% divide_eq_imp
tff(fact_2562_divide__eq__imp,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B2 = aa_rat_rat(times_times_rat(A2),C) )
       => ( divide_divide_rat(B2,C) = A2 ) ) ) ).

% divide_eq_imp
tff(fact_2563_eq__divide__imp,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(A2),C) = B2 )
       => ( A2 = aa_real_real(divide_divide_real(B2),C) ) ) ) ).

% eq_divide_imp
tff(fact_2564_eq__divide__imp,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A2),C) = B2 )
       => ( A2 = divide_divide_rat(B2,C) ) ) ) ).

% eq_divide_imp
tff(fact_2565_nonzero__divide__eq__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(divide_divide_real(B2),C) = A2 )
      <=> ( B2 = aa_real_real(times_times_real(A2),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2566_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat(B2,C) = A2 )
      <=> ( B2 = aa_rat_rat(times_times_rat(A2),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2567_nonzero__eq__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( A2 = aa_real_real(divide_divide_real(B2),C) )
      <=> ( aa_real_real(times_times_real(A2),C) = B2 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2568_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A2 = divide_divide_rat(B2,C) )
      <=> ( aa_rat_rat(times_times_rat(A2),C) = B2 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2569_eq__add__iff1,axiom,
    ! [A2: real,E2: real,C: real,B2: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A2),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(B2),E2)),D) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(minus_minus_real(A2),B2)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_2570_eq__add__iff1,axiom,
    ! [A2: rat,E2: rat,C: rat,B2: rat,D: rat] :
      ( ( plus_plus_rat(aa_rat_rat(times_times_rat(A2),E2),C) = plus_plus_rat(aa_rat_rat(times_times_rat(B2),E2),D) )
    <=> ( plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A2,B2)),E2),C) = D ) ) ).

% eq_add_iff1
tff(fact_2571_eq__add__iff1,axiom,
    ! [A2: int,E2: int,C: int,B2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),D) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_2572_eq__add__iff2,axiom,
    ! [A2: real,E2: real,C: real,B2: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A2),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(B2),E2)),D) )
    <=> ( C = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(minus_minus_real(B2),A2)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_2573_eq__add__iff2,axiom,
    ! [A2: rat,E2: rat,C: rat,B2: rat,D: rat] :
      ( ( plus_plus_rat(aa_rat_rat(times_times_rat(A2),E2),C) = plus_plus_rat(aa_rat_rat(times_times_rat(B2),E2),D) )
    <=> ( C = plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B2,A2)),E2),D) ) ) ).

% eq_add_iff2
tff(fact_2574_eq__add__iff2,axiom,
    ! [A2: int,E2: int,C: int,B2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),D) )
    <=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A2)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_2575_square__diff__square__factored,axiom,
    ! [Xb: real,Y: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y),Y)) = aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_real_real(minus_minus_real(Xb),Y)) ).

% square_diff_square_factored
tff(fact_2576_square__diff__square__factored,axiom,
    ! [Xb: rat,Y: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Xb),aa_rat_rat(times_times_rat(Y),Y)) = aa_rat_rat(times_times_rat(plus_plus_rat(Xb,Y)),minus_minus_rat(Xb,Y)) ).

% square_diff_square_factored
tff(fact_2577_square__diff__square__factored,axiom,
    ! [Xb: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)) ).

% square_diff_square_factored
tff(fact_2578_left__right__inverse__power,axiom,
    ! [Xb: uint32,Y: uint32,N: nat] :
      ( ( aa_uint32_uint32(times_times_uint32(Xb),Y) = one_one_uint32 )
     => ( aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(Xb),N)),aa_nat_uint32(power_power_uint32(Y),N)) = one_one_uint32 ) ) ).

% left_right_inverse_power
tff(fact_2579_left__right__inverse__power,axiom,
    ! [Xb: complex,Y: complex,N: nat] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y) = one_one_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y),N)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_2580_left__right__inverse__power,axiom,
    ! [Xb: code_integer,Y: code_integer,N: nat] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),Y) = one_one_Code_integer )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),N)) = one_one_Code_integer ) ) ).

% left_right_inverse_power
tff(fact_2581_left__right__inverse__power,axiom,
    ! [Xb: real,Y: real,N: nat] :
      ( ( aa_real_real(times_times_real(Xb),Y) = one_one_real )
     => ( aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_2582_left__right__inverse__power,axiom,
    ! [Xb: rat,Y: rat,N: nat] :
      ( ( aa_rat_rat(times_times_rat(Xb),Y) = one_one_rat )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),N)),aa_nat_rat(power_power_rat(Y),N)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_2583_left__right__inverse__power,axiom,
    ! [Xb: nat,Y: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_2584_left__right__inverse__power,axiom,
    ! [Xb: int,Y: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),N)),aa_nat_int(power_power_int(Y),N)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_2585_power__Suc2,axiom,
    ! [A2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(suc,N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)),A2) ).

% power_Suc2
tff(fact_2586_power__Suc2,axiom,
    ! [A2: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),A2) ).

% power_Suc2
tff(fact_2587_power__Suc2,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),A2) ).

% power_Suc2
tff(fact_2588_power__Suc2,axiom,
    ! [A2: rat,N: nat] : aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),N)),A2) ).

% power_Suc2
tff(fact_2589_power__Suc2,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),A2) ).

% power_Suc2
tff(fact_2590_power__Suc2,axiom,
    ! [A2: int,N: nat] : aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A2),N)),A2) ).

% power_Suc2
tff(fact_2591_power__Suc,axiom,
    ! [A2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(suc,N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)) ).

% power_Suc
tff(fact_2592_power__Suc,axiom,
    ! [A2: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ).

% power_Suc
tff(fact_2593_power__Suc,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,N)) = aa_real_real(times_times_real(A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ).

% power_Suc
tff(fact_2594_power__Suc,axiom,
    ! [A2: rat,N: nat] : aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,N)) = aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(A2),N)) ).

% power_Suc
tff(fact_2595_power__Suc,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ).

% power_Suc
tff(fact_2596_power__Suc,axiom,
    ! [A2: int,N: nat] : aa_nat_int(power_power_int(A2),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(power_power_int(A2),N)) ).

% power_Suc
tff(fact_2597_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% Suc_mult_less_cancel1
tff(fact_2598_power__add,axiom,
    ! [A2: complex,M: nat,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),M)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)) ).

% power_add
tff(fact_2599_power__add,axiom,
    ! [A2: code_integer,M: nat,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ).

% power_add
tff(fact_2600_power__add,axiom,
    ! [A2: real,M: nat,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ).

% power_add
tff(fact_2601_power__add,axiom,
    ! [A2: rat,M: nat,N: nat] : aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),M)),aa_nat_rat(power_power_rat(A2),N)) ).

% power_add
tff(fact_2602_power__add,axiom,
    ! [A2: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ).

% power_add
tff(fact_2603_power__add,axiom,
    ! [A2: int,M: nat,N: nat] : aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),N)) ).

% power_add
tff(fact_2604_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N) )
      <=> ( M = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_2605_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ) ).

% nat_mult_less_cancel1
tff(fact_2606_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ) ).

% mult_less_mono1
tff(fact_2607_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ) ).

% mult_less_mono2
tff(fact_2608_mult__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ).

% mult_Suc
tff(fact_2609_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% Suc_mult_le_cancel1
tff(fact_2610_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_2611_mlex__snd__decrI,axiom,
    ! [A2: nat,A4: nat,B2: nat,B3: nat,N2: nat] :
      ( ( A2 = A4 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N2)),B3)) ) ) ).

% mlex_snd_decrI
tff(fact_2612_mlex__fst__decrI,axiom,
    ! [A2: nat,A4: nat,B2: nat,N2: nat,B3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),N2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B3),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N2)),B3)) ) ) ) ).

% mlex_fst_decrI
tff(fact_2613_mlex__bound,axiom,
    ! [A2: nat,A3: nat,B2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),N2)) ) ) ).

% mlex_bound
tff(fact_2614_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),N))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),I) ) ).

% less_mult_imp_div_less
tff(fact_2615_mlex__leI,axiom,
    ! [A2: nat,A4: nat,B2: nat,B3: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),B3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N2)),B3)) ) ) ).

% mlex_leI
tff(fact_2616_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)),M) ).

% div_times_less_eq_dividend
tff(fact_2617_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))),M) ).

% times_div_less_eq_dividend
tff(fact_2618_div__mult__le,axiom,
    ! [A2: nat,B2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)),A2) ).

% div_mult_le
tff(fact_2619_odd__nonzero,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z) != zero_zero_int ).

% odd_nonzero
tff(fact_2620_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I)
     => ( aa_int_o(P,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_ge_induct
tff(fact_2621_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_gr_induct
tff(fact_2622_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_2623_power__odd__eq,axiom,
    ! [A2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2624_power__odd__eq,axiom,
    ! [A2: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2625_power__odd__eq,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_real_real(times_times_real(A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2626_power__odd__eq,axiom,
    ! [A2: rat,N: nat] : aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2627_power__odd__eq,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2628_power__odd__eq,axiom,
    ! [A2: int,N: nat] : aa_nat_int(power_power_int(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(power_power_int(aa_nat_int(power_power_int(A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2629_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,Xb: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),Y)
     => ( plus_p3455044024723400733d_enat(Xb,minus_3235023915231533773d_enat(Y,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(Xb,Y),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_2630_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% Suc_double_not_eq_double
tff(fact_2631_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% double_not_eq_Suc_double
tff(fact_2632_mult__le__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ) ) ).

% mult_le_cancel_left
tff(fact_2633_mult__le__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A2)),aa_rat_rat(times_times_rat(C),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ) ) ).

% mult_le_cancel_left
tff(fact_2634_mult__le__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ) ) ).

% mult_le_cancel_left
tff(fact_2635_mult__le__cancel__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ) ) ).

% mult_le_cancel_right
tff(fact_2636_mult__le__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ) ) ).

% mult_le_cancel_right
tff(fact_2637_mult__le__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ) ) ).

% mult_le_cancel_right
tff(fact_2638_mult__left__less__imp__less,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% mult_left_less_imp_less
tff(fact_2639_mult__left__less__imp__less,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A2)),aa_rat_rat(times_times_rat(C),B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ) ).

% mult_left_less_imp_less
tff(fact_2640_mult__left__less__imp__less,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ) ).

% mult_left_less_imp_less
tff(fact_2641_mult__left__less__imp__less,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ) ).

% mult_left_less_imp_less
tff(fact_2642_mult__strict__mono,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2643_mult__strict__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2644_mult__strict__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2645_mult__strict__mono,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2646_mult__less__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ) ) ).

% mult_less_cancel_left
tff(fact_2647_mult__less__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A2)),aa_rat_rat(times_times_rat(C),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ) ) ).

% mult_less_cancel_left
tff(fact_2648_mult__less__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2) ) ) ) ).

% mult_less_cancel_left
tff(fact_2649_mult__right__less__imp__less,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% mult_right_less_imp_less
tff(fact_2650_mult__right__less__imp__less,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ) ).

% mult_right_less_imp_less
tff(fact_2651_mult__right__less__imp__less,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ) ).

% mult_right_less_imp_less
tff(fact_2652_mult__right__less__imp__less,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ) ).

% mult_right_less_imp_less
tff(fact_2653_mult__strict__mono_H,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2654_mult__strict__mono_H,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2655_mult__strict__mono_H,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2656_mult__strict__mono_H,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2657_mult__less__cancel__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ) ) ).

% mult_less_cancel_right
tff(fact_2658_mult__less__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ) ) ).

% mult_less_cancel_right
tff(fact_2659_mult__less__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2) ) ) ) ).

% mult_less_cancel_right
tff(fact_2660_mult__le__cancel__left__neg,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2661_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A2)),aa_rat_rat(times_times_rat(C),B2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2662_mult__le__cancel__left__neg,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2663_mult__le__cancel__left__pos,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2664_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A2)),aa_rat_rat(times_times_rat(C),B2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2665_mult__le__cancel__left__pos,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2666_mult__left__le__imp__le,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ).

% mult_left_le_imp_le
tff(fact_2667_mult__left__le__imp__le,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A2)),aa_rat_rat(times_times_rat(C),B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ) ).

% mult_left_le_imp_le
tff(fact_2668_mult__left__le__imp__le,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ) ).

% mult_left_le_imp_le
tff(fact_2669_mult__left__le__imp__le,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ) ).

% mult_left_le_imp_le
tff(fact_2670_mult__right__le__imp__le,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ).

% mult_right_le_imp_le
tff(fact_2671_mult__right__le__imp__le,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ) ).

% mult_right_le_imp_le
tff(fact_2672_mult__right__le__imp__le,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ) ).

% mult_right_le_imp_le
tff(fact_2673_mult__right__le__imp__le,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ) ).

% mult_right_le_imp_le
tff(fact_2674_mult__le__less__imp__less,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2675_mult__le__less__imp__less,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2676_mult__le__less__imp__less,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2677_mult__le__less__imp__less,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2678_mult__less__le__imp__less,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2679_mult__less__le__imp__less,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2680_mult__less__le__imp__less,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2681_mult__less__le__imp__less,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2682_sum__squares__le__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y),Y))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2683_sum__squares__le__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat)
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2684_sum__squares__le__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2685_sum__squares__ge__zero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2686_sum__squares__ge__zero,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb),aa_rat_rat(times_times_rat(Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2687_sum__squares__ge__zero,axiom,
    ! [Xb: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2688_mult__left__le__one__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Y),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_2689_mult__left__le__one__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Y),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_2690_mult__left__le__one__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_2691_mult__right__le__one__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_2692_mult__right__le__one__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_2693_mult__right__le__one__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_2694_mult__le__one,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),B2)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_2695_mult__le__one,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),B2)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_2696_mult__le__one,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_2697_mult__le__one,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_2698_mult__left__le,axiom,
    ! [C: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),C)),A2) ) ) ).

% mult_left_le
tff(fact_2699_mult__left__le,axiom,
    ! [C: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C)),A2) ) ) ).

% mult_left_le
tff(fact_2700_mult__left__le,axiom,
    ! [C: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),A2) ) ) ).

% mult_left_le
tff(fact_2701_mult__left__le,axiom,
    ! [C: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),A2) ) ) ).

% mult_left_le
tff(fact_2702_sum__squares__gt__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y),Y)))
    <=> ( ( Xb != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2703_sum__squares__gt__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb),aa_rat_rat(times_times_rat(Y),Y)))
    <=> ( ( Xb != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2704_sum__squares__gt__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))
    <=> ( ( Xb != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2705_not__sum__squares__lt__zero,axiom,
    ! [Xb: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Y),Y))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_2706_not__sum__squares__lt__zero,axiom,
    ! [Xb: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb),aa_rat_rat(times_times_rat(Y),Y))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_2707_not__sum__squares__lt__zero,axiom,
    ! [Xb: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_2708_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2709_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2710_divide__less__eq,axiom,
    ! [B2: real,C: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B2),C)),A2)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(times_times_real(A2),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),B2),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)) ) ) ).

% divide_less_eq
tff(fact_2711_divide__less__eq,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B2,C)),A2)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)) ) ) ).

% divide_less_eq
tff(fact_2712_less__divide__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(divide_divide_real(B2),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),B2),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(times_times_real(A2),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)) ) ) ).

% less_divide_eq
tff(fact_2713_less__divide__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),divide_divide_rat(B2,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),B2),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_2714_neg__divide__less__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B2),C)),A2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),B2) ) ) ).

% neg_divide_less_eq
tff(fact_2715_neg__divide__less__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B2,C)),A2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),B2) ) ) ).

% neg_divide_less_eq
tff(fact_2716_neg__less__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(divide_divide_real(B2),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(times_times_real(A2),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2717_neg__less__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),divide_divide_rat(B2,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2718_pos__divide__less__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B2),C)),A2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(times_times_real(A2),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2719_pos__divide__less__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B2,C)),A2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2720_pos__less__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(divide_divide_real(B2),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),B2) ) ) ).

% pos_less_divide_eq
tff(fact_2721_pos__less__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),divide_divide_rat(B2,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),B2) ) ) ).

% pos_less_divide_eq
tff(fact_2722_mult__imp__div__pos__less,axiom,
    ! [Y: real,Xb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(times_times_real(Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xb),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_2723_mult__imp__div__pos__less,axiom,
    ! [Y: rat,Xb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(times_times_rat(Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xb,Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_2724_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Z),Y)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(divide_divide_real(Xb),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2725_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Z),Y)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),divide_divide_rat(Xb,Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2726_divide__strict__left__mono,axiom,
    ! [B2: real,A2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A2),B2))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(C),A2)),aa_real_real(divide_divide_real(C),B2)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2727_divide__strict__left__mono,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(C,A2)),divide_divide_rat(C,B2)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2728_divide__strict__left__mono__neg,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A2),B2))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(C),A2)),aa_real_real(divide_divide_real(C),B2)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2729_divide__strict__left__mono__neg,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(C,A2)),divide_divide_rat(C,B2)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2730_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( ( aa_real_real(divide_divide_real(B2),C) = numeral_numeral_real(W) )
    <=> $ite(C != zero_zero_real,B2 = aa_real_real(times_times_real(numeral_numeral_real(W)),C),numeral_numeral_real(W) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2731_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( ( divide_divide_rat(B2,C) = numeral_numeral_rat(W) )
    <=> $ite(C != zero_zero_rat,B2 = aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C),numeral_numeral_rat(W) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2732_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( ( numeral_numeral_real(W) = aa_real_real(divide_divide_real(B2),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(numeral_numeral_real(W)),C) = B2,numeral_numeral_real(W) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2733_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( ( numeral_numeral_rat(W) = divide_divide_rat(B2,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C) = B2,numeral_numeral_rat(W) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2734_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: real,E2: real,C: real,B2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A2),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(B2),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(minus_minus_real(B2),A2)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_2735_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: rat,E2: rat,C: rat,B2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A2),E2),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B2),E2),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B2,A2)),E2),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_2736_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: int,E2: int,C: int,B2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A2)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_2737_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: real,E2: real,C: real,B2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A2),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(B2),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(minus_minus_real(A2),B2)),E2)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_2738_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: rat,E2: rat,C: rat,B2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A2),E2),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B2),E2),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A2,B2)),E2),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_2739_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: int,E2: int,C: int,B2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),E2)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_2740_less__add__iff1,axiom,
    ! [A2: real,E2: real,C: real,B2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A2),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(B2),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(minus_minus_real(A2),B2)),E2)),C)),D) ) ).

% less_add_iff1
tff(fact_2741_less__add__iff1,axiom,
    ! [A2: rat,E2: rat,C: rat,B2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A2),E2),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B2),E2),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(A2,B2)),E2),C)),D) ) ).

% less_add_iff1
tff(fact_2742_less__add__iff1,axiom,
    ! [A2: int,E2: int,C: int,B2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),E2)),C)),D) ) ).

% less_add_iff1
tff(fact_2743_less__add__iff2,axiom,
    ! [A2: real,E2: real,C: real,B2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A2),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(B2),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(minus_minus_real(B2),A2)),E2)),D)) ) ).

% less_add_iff2
tff(fact_2744_less__add__iff2,axiom,
    ! [A2: rat,E2: rat,C: rat,B2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(A2),E2),C)),plus_plus_rat(aa_rat_rat(times_times_rat(B2),E2),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),plus_plus_rat(aa_rat_rat(times_times_rat(minus_minus_rat(B2,A2)),E2),D)) ) ).

% less_add_iff2
tff(fact_2745_less__add__iff2,axiom,
    ! [A2: int,E2: int,C: int,B2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E2)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A2)),E2)),D)) ) ).

% less_add_iff2
tff(fact_2746_add__divide__eq__if__simps_I2_J,axiom,
    ! [A2: real,Z: real,B2: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(A2),Z)),B2) = $ite(Z = zero_zero_real,B2,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(times_times_real(B2),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2747_add__divide__eq__if__simps_I2_J,axiom,
    ! [A2: rat,Z: rat,B2: rat] :
      plus_plus_rat(divide_divide_rat(A2,Z),B2) = $ite(Z = zero_zero_rat,B2,divide_divide_rat(plus_plus_rat(A2,aa_rat_rat(times_times_rat(B2),Z)),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2748_add__divide__eq__if__simps_I1_J,axiom,
    ! [A2: real,B2: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(divide_divide_real(B2),Z)) = $ite(Z = zero_zero_real,A2,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(A2),Z)),B2)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2749_add__divide__eq__if__simps_I1_J,axiom,
    ! [A2: rat,B2: rat,Z: rat] :
      plus_plus_rat(A2,divide_divide_rat(B2,Z)) = $ite(Z = zero_zero_rat,A2,divide_divide_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),Z),B2),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2750_add__frac__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(Xb),Y)),aa_real_real(divide_divide_real(W),Z)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(W),Y))),aa_real_real(times_times_real(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2751_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( plus_plus_rat(divide_divide_rat(Xb,Y),divide_divide_rat(W,Z)) = divide_divide_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2752_add__frac__num,axiom,
    ! [Y: real,Xb: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(Xb),Y)),Z) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(times_times_real(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2753_add__frac__num,axiom,
    ! [Y: rat,Xb: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( plus_plus_rat(divide_divide_rat(Xb,Y),Z) = divide_divide_rat(plus_plus_rat(Xb,aa_rat_rat(times_times_rat(Z),Y)),Y) ) ) ).

% add_frac_num
tff(fact_2754_add__num__frac,axiom,
    ! [Y: real,Z: real,Xb: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(divide_divide_real(Xb),Y)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(times_times_real(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2755_add__num__frac,axiom,
    ! [Y: rat,Z: rat,Xb: rat] :
      ( ( Y != zero_zero_rat )
     => ( plus_plus_rat(Z,divide_divide_rat(Xb,Y)) = divide_divide_rat(plus_plus_rat(Xb,aa_rat_rat(times_times_rat(Z),Y)),Y) ) ) ).

% add_num_frac
tff(fact_2756_add__divide__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(divide_divide_real(Y),Z)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(Xb),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2757_add__divide__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( plus_plus_rat(Xb,divide_divide_rat(Y,Z)) = divide_divide_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Z),Y),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2758_divide__add__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(Xb),Z)),Y) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(times_times_real(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2759_divide__add__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( plus_plus_rat(divide_divide_rat(Xb,Z),Y) = divide_divide_rat(plus_plus_rat(Xb,aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2760_add__divide__eq__if__simps_I4_J,axiom,
    ! [A2: real,B2: real,Z: real] :
      aa_real_real(minus_minus_real(A2),aa_real_real(divide_divide_real(B2),Z)) = $ite(Z = zero_zero_real,A2,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(A2),Z)),B2)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_2761_add__divide__eq__if__simps_I4_J,axiom,
    ! [A2: rat,B2: rat,Z: rat] :
      minus_minus_rat(A2,divide_divide_rat(B2,Z)) = $ite(Z = zero_zero_rat,A2,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A2),Z),B2),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_2762_diff__frac__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Xb),Y)),aa_real_real(divide_divide_real(W),Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(W),Y))),aa_real_real(times_times_real(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2763_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( minus_minus_rat(divide_divide_rat(Xb,Y),divide_divide_rat(W,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2764_diff__divide__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(Xb),aa_real_real(divide_divide_real(Y),Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_2765_diff__divide__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(Xb,divide_divide_rat(Y,Z)) = divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z),Y),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_2766_divide__diff__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Xb),Z)),Y) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Xb),aa_real_real(times_times_real(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_2767_divide__diff__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(divide_divide_rat(Xb,Z),Y) = divide_divide_rat(minus_minus_rat(Xb,aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_2768_square__diff__one__factored,axiom,
    ! [Xb: uint32] : minus_minus_uint32(aa_uint32_uint32(times_times_uint32(Xb),Xb),one_one_uint32) = aa_uint32_uint32(times_times_uint32(plus_plus_uint32(Xb,one_one_uint32)),minus_minus_uint32(Xb,one_one_uint32)) ).

% square_diff_one_factored
tff(fact_2769_square__diff__one__factored,axiom,
    ! [Xb: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Xb)),one_one_real) = aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)),aa_real_real(minus_minus_real(Xb),one_one_real)) ).

% square_diff_one_factored
tff(fact_2770_square__diff__one__factored,axiom,
    ! [Xb: rat] : minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Xb),one_one_rat) = aa_rat_rat(times_times_rat(plus_plus_rat(Xb,one_one_rat)),minus_minus_rat(Xb,one_one_rat)) ).

% square_diff_one_factored
tff(fact_2771_square__diff__one__factored,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),one_one_int)) ).

% square_diff_one_factored
tff(fact_2772_power__gt1__lemma,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))) ) ).

% power_gt1_lemma
tff(fact_2773_power__gt1__lemma,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(times_times_real(A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ).

% power_gt1_lemma
tff(fact_2774_power__gt1__lemma,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(A2),N))) ) ).

% power_gt1_lemma
tff(fact_2775_power__gt1__lemma,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ).

% power_gt1_lemma
tff(fact_2776_power__gt1__lemma,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(power_power_int(A2),N))) ) ).

% power_gt1_lemma
tff(fact_2777_power__less__power__Suc,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))) ) ).

% power_less_power_Suc
tff(fact_2778_power__less__power__Suc,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_real_real(times_times_real(A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ) ).

% power_less_power_Suc
tff(fact_2779_power__less__power__Suc,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),N)),aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(A2),N))) ) ).

% power_less_power_Suc
tff(fact_2780_power__less__power__Suc,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))) ) ).

% power_less_power_Suc
tff(fact_2781_power__less__power__Suc,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),N)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(power_power_int(A2),N))) ) ).

% power_less_power_Suc
tff(fact_2782_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ) ) ).

% one_less_mult
tff(fact_2783_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) ) ) ).

% n_less_m_mult_n
tff(fact_2784_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)) ) ) ).

% n_less_n_mult_m
tff(fact_2785_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% nat_mult_le_cancel1
tff(fact_2786_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_2787_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q2)),N)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_2788_td__gal__lt,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C)),B2) ) ) ).

% td_gal_lt
tff(fact_2789_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_2790_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_2791_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N)) ) ) ).

% nat_le_add_iff2
tff(fact_2792_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N) ) ) ).

% nat_le_add_iff1
tff(fact_2793_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N) )
      <=> ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_2794_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_2795_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)),Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% odd_less_0_iff
tff(fact_2796_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% add1_zle_eq
tff(fact_2797_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z) ) ).

% zless_imp_add1_zle
tff(fact_2798_int__induct,axiom,
    ! [P: fun_int_o,K: int,I: int] :
      ( aa_int_o(P,K)
     => ( ! [I2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
           => ( aa_int_o(P,I2)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_induct
tff(fact_2799_power__numeral__even,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: complex,
        w:= aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,w),w) ) ).

% power_numeral_even
tff(fact_2800_power__numeral__even,axiom,
    ! [Z: code_integer,W: num] :
      aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: code_integer,
        w:= aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,w),w) ) ).

% power_numeral_even
tff(fact_2801_power__numeral__even,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(times_times_real(w),w) ) ).

% power_numeral_even
tff(fact_2802_power__numeral__even,axiom,
    ! [Z: rat,W: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(times_times_rat(w),w) ) ).

% power_numeral_even
tff(fact_2803_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,w),w) ) ).

% power_numeral_even
tff(fact_2804_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: int,
        w:= aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,w),w) ) ).

% power_numeral_even
tff(fact_2805_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,
    ! [A2: $o,B2: $o,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,vEBT_Leaf((A2),(B2))),Xb) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_2806_field__le__mult__one__interval,axiom,
    ! [Xb: real,Y: real] :
      ( ! [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Z2),Xb)),Y) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% field_le_mult_one_interval
tff(fact_2807_field__le__mult__one__interval,axiom,
    ! [Xb: rat,Y: rat] :
      ( ! [Z2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),one_one_rat)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z2),Xb)),Y) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% field_le_mult_one_interval
tff(fact_2808_mult__le__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(times_times_real(C),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),one_one_real) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2809_mult__le__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(C),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2810_mult__le__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2811_mult__le__cancel__left2,axiom,
    ! [C: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A2)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A2) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2812_mult__le__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A2)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A2) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2813_mult__le__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A2) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2814_mult__le__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(times_times_real(B2),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),one_one_real) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2815_mult__le__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(B2),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2816_mult__le__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2817_mult__le__cancel__right2,axiom,
    ! [A2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A2) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2818_mult__le__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A2) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2819_mult__le__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A2) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2820_mult__less__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(times_times_real(C),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),one_one_real) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2821_mult__less__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(C),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2822_mult__less__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2823_mult__less__cancel__left2,axiom,
    ! [C: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A2)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2824_mult__less__cancel__left2,axiom,
    ! [C: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A2)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2825_mult__less__cancel__left2,axiom,
    ! [C: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2826_mult__less__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(times_times_real(B2),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),one_one_real) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2827_mult__less__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(B2),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2828_mult__less__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2829_mult__less__cancel__right2,axiom,
    ! [A2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2830_mult__less__cancel__right2,axiom,
    ! [A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2831_mult__less__cancel__right2,axiom,
    ! [A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2832_convex__bound__le,axiom,
    ! [Xb: real,A2: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(U),Xb)),aa_real_real(times_times_real(V),Y))),A2) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2833_convex__bound__le,axiom,
    ! [Xb: rat,A2: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( plus_plus_rat(U,V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(aa_rat_rat(times_times_rat(U),Xb),aa_rat_rat(times_times_rat(V),Y))),A2) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2834_convex__bound__le,axiom,
    ! [Xb: int,A2: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A2) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2835_divide__le__eq,axiom,
    ! [B2: real,C: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B2),C)),A2)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(times_times_real(A2),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),C)),B2),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)) ) ) ).

% divide_le_eq
tff(fact_2836_divide__le__eq,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B2,C)),A2)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)) ) ) ).

% divide_le_eq
tff(fact_2837_le__divide__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(divide_divide_real(B2),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),C)),B2),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(times_times_real(A2),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)) ) ) ).

% le_divide_eq
tff(fact_2838_le__divide__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),divide_divide_rat(B2,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C)),B2),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_2839_divide__left__mono,axiom,
    ! [B2: real,A2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A2),B2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(C),A2)),aa_real_real(divide_divide_real(C),B2)) ) ) ) ).

% divide_left_mono
tff(fact_2840_divide__left__mono,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(C,A2)),divide_divide_rat(C,B2)) ) ) ) ).

% divide_left_mono
tff(fact_2841_neg__divide__le__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B2),C)),A2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),C)),B2) ) ) ).

% neg_divide_le_eq
tff(fact_2842_neg__divide__le__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B2,C)),A2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C)),B2) ) ) ).

% neg_divide_le_eq
tff(fact_2843_neg__le__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(divide_divide_real(B2),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(times_times_real(A2),C)) ) ) ).

% neg_le_divide_eq
tff(fact_2844_neg__le__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),divide_divide_rat(B2,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),C)) ) ) ).

% neg_le_divide_eq
tff(fact_2845_pos__divide__le__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B2),C)),A2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(times_times_real(A2),C)) ) ) ).

% pos_divide_le_eq
tff(fact_2846_pos__divide__le__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B2,C)),A2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),C)) ) ) ).

% pos_divide_le_eq
tff(fact_2847_pos__le__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(divide_divide_real(B2),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),C)),B2) ) ) ).

% pos_le_divide_eq
tff(fact_2848_pos__le__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),divide_divide_rat(B2,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C)),B2) ) ) ).

% pos_le_divide_eq
tff(fact_2849_mult__imp__div__pos__le,axiom,
    ! [Y: real,Xb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(times_times_real(Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_2850_mult__imp__div__pos__le,axiom,
    ! [Y: rat,Xb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(times_times_rat(Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xb,Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_2851_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Z),Y)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(divide_divide_real(Xb),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2852_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Y)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),divide_divide_rat(Xb,Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2853_divide__left__mono__neg,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A2),B2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(C),A2)),aa_real_real(divide_divide_real(C),B2)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2854_divide__left__mono__neg,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(C,A2)),divide_divide_rat(C,B2)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2855_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B2),C)),numeral_numeral_real(W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(times_times_real(numeral_numeral_real(W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(numeral_numeral_real(W)),C)),B2),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),numeral_numeral_real(W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2856_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B2,C)),numeral_numeral_rat(W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2857_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W)),aa_real_real(divide_divide_real(B2),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(numeral_numeral_real(W)),C)),B2),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(times_times_real(numeral_numeral_real(W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2858_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(W)),divide_divide_rat(B2,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B2),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2859_mult__2,axiom,
    ! [Z: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Z) = plus_p361126936061061375l_num1(Z,Z) ).

% mult_2
tff(fact_2860_mult__2,axiom,
    ! [Z: real] : aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),Z) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2
tff(fact_2861_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Z) = plus_plus_rat(Z,Z) ).

% mult_2
tff(fact_2862_mult__2,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2
tff(fact_2863_mult__2,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Z) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2
tff(fact_2864_mult__2__right,axiom,
    ! [Z: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(Z),numera7442385471795722001l_num1(aa_num_num(bit0,one))) = plus_p361126936061061375l_num1(Z,Z) ).

% mult_2_right
tff(fact_2865_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(times_times_real(Z),numeral_numeral_real(aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_2866_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),numeral_numeral_rat(aa_num_num(bit0,one))) = plus_plus_rat(Z,Z) ).

% mult_2_right
tff(fact_2867_mult__2__right,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2_right
tff(fact_2868_mult__2__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2_right
tff(fact_2869_left__add__twice,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(A2,plus_p361126936061061375l_num1(A2,B2)) = plus_p361126936061061375l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2),B2) ).

% left_add_twice
tff(fact_2870_left__add__twice,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),A2)),B2) ).

% left_add_twice
tff(fact_2871_left__add__twice,axiom,
    ! [A2: rat,B2: rat] : plus_plus_rat(A2,plus_plus_rat(A2,B2)) = plus_plus_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),A2),B2) ).

% left_add_twice
tff(fact_2872_left__add__twice,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)),B2) ).

% left_add_twice
tff(fact_2873_left__add__twice,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)),B2) ).

% left_add_twice
tff(fact_2874_frac__le__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Y)),aa_real_real(divide_divide_real(W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(W),Y))),aa_real_real(times_times_real(Y),Z))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_2875_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(Xb,Y)),divide_divide_rat(W,Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_2876_power__Suc__less,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ) ) ).

% power_Suc_less
tff(fact_2877_power__Suc__less,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ) ) ).

% power_Suc_less
tff(fact_2878_power__Suc__less,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(A2),N))),aa_nat_rat(power_power_rat(A2),N)) ) ) ).

% power_Suc_less
tff(fact_2879_power__Suc__less,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ) ) ).

% power_Suc_less
tff(fact_2880_power__Suc__less,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(power_power_int(A2),N))),aa_nat_int(power_power_int(A2),N)) ) ) ).

% power_Suc_less
tff(fact_2881_frac__less__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xb),Y)),aa_real_real(divide_divide_real(W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(W),Y))),aa_real_real(times_times_real(Y),Z))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_2882_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(Xb,Y)),divide_divide_rat(W,Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z),aa_rat_rat(times_times_rat(W),Y)),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_2883_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),one))),N) ).

% Suc_nat_number_of_add
tff(fact_2884_power4__eq__xxxx,axiom,
    ! [Xb: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_2885_power4__eq__xxxx,axiom,
    ! [Xb: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_2886_power4__eq__xxxx,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_2887_power4__eq__xxxx,axiom,
    ! [Xb: rat] : aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_2888_power4__eq__xxxx,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_2889_power4__eq__xxxx,axiom,
    ! [Xb: int] : aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_2890_power2__eq__square,axiom,
    ! [A2: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),A2) ).

% power2_eq_square
tff(fact_2891_power2__eq__square,axiom,
    ! [A2: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),A2) ).

% power2_eq_square
tff(fact_2892_power2__eq__square,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(times_times_real(A2),A2) ).

% power2_eq_square
tff(fact_2893_power2__eq__square,axiom,
    ! [A2: rat] : aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(A2),A2) ).

% power2_eq_square
tff(fact_2894_power2__eq__square,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),A2) ).

% power2_eq_square
tff(fact_2895_power2__eq__square,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),A2) ).

% power2_eq_square
tff(fact_2896_power__even__eq,axiom,
    ! [A2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2897_power__even__eq,axiom,
    ! [A2: int,N: nat] : aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2898_power__even__eq,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2899_power__even__eq,axiom,
    ! [A2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2900_power__even__eq,axiom,
    ! [A2: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2901_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q2)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_2902_split__div,axiom,
    ! [P: fun_nat_o,M: nat,N: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))
    <=> ( ( ( N = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),N)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I4)),J3) )
               => aa_nat_o(P,I4) ) ) ) ) ) ).

% split_div
tff(fact_2903_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N))) ) ).

% dividend_less_div_times
tff(fact_2904_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)))) ) ).

% dividend_less_times_div
tff(fact_2905_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),Q2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),N) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_2906_td__gal,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)),A2)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C)) ) ) ).

% td_gal
tff(fact_2907_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N) = $ite(M = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),N))) ).

% mult_eq_if
tff(fact_2908_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),N) ) ) ).

% nat_less_add_iff1
tff(fact_2909_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),N))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),N)) ) ) ).

% nat_less_add_iff2
tff(fact_2910_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ).

% le_imp_0_less
tff(fact_2911_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Va: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,vEBT_Leaf((A2),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
tff(fact_2912_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,vEBT_Leaf((Uu),(Uv2))),zero_zero_nat) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
tff(fact_2913_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,N: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,vEBT_Leaf((Uv2),(Uw2))),aa_nat_nat(suc,N)) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
tff(fact_2914_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] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va)),Vb) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
tff(fact_2915_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,vEBT_Leaf((Uu),(B2))),zero_zero_nat) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
tff(fact_2916_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] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)),Va) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
tff(fact_2917_convex__bound__lt,axiom,
    ! [Xb: real,A2: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(U),Xb)),aa_real_real(times_times_real(V),Y))),A2) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2918_convex__bound__lt,axiom,
    ! [Xb: rat,A2: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( plus_plus_rat(U,V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,plus_plus_rat(aa_rat_rat(times_times_rat(U),Xb),aa_rat_rat(times_times_rat(V),Y))),A2) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2919_convex__bound__lt,axiom,
    ! [Xb: int,A2: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A2) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2920_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B2),C)),numeral_numeral_real(W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(times_times_real(numeral_numeral_real(W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(numeral_numeral_real(W)),C)),B2),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),numeral_numeral_real(W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2921_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B2,C)),numeral_numeral_rat(W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),numeral_numeral_rat(W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2922_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(W)),aa_real_real(divide_divide_real(B2),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(numeral_numeral_real(W)),C)),B2),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(times_times_real(numeral_numeral_real(W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(W)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2923_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(W)),divide_divide_rat(B2,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),B2),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2924_scaling__mono,axiom,
    ! [U: real,V: real,R: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),V)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R),S)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),aa_real_real(divide_divide_real(aa_real_real(times_times_real(R),aa_real_real(minus_minus_real(V),U))),S))),V) ) ) ) ).

% scaling_mono
tff(fact_2925_scaling__mono,axiom,
    ! [U: rat,V: rat,R: rat,S: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),V)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,plus_plus_rat(U,divide_divide_rat(aa_rat_rat(times_times_rat(R),minus_minus_rat(V,U)),S))),V) ) ) ) ).

% scaling_mono
tff(fact_2926_power__eq__if,axiom,
    ! [P2: uint32,M: nat] :
      aa_nat_uint32(power_power_uint32(P2),M) = $ite(M = zero_zero_nat,one_one_uint32,aa_uint32_uint32(times_times_uint32(P2),aa_nat_uint32(power_power_uint32(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_2927_power__eq__if,axiom,
    ! [P2: complex,M: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,P2),M) = $ite(M = zero_zero_nat,one_one_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,P2),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_2928_power__eq__if,axiom,
    ! [P2: code_integer,M: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(P2),M) = $ite(M = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,P2),aa_nat_Code_integer(power_8256067586552552935nteger(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_2929_power__eq__if,axiom,
    ! [P2: real,M: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),M) = $ite(M = zero_zero_nat,one_one_real,aa_real_real(times_times_real(P2),aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_2930_power__eq__if,axiom,
    ! [P2: rat,M: nat] :
      aa_nat_rat(power_power_rat(P2),M) = $ite(M = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(P2),aa_nat_rat(power_power_rat(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_2931_power__eq__if,axiom,
    ! [P2: nat,M: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),M) = $ite(M = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_2932_power__eq__if,axiom,
    ! [P2: int,M: nat] :
      aa_nat_int(power_power_int(P2),M) = $ite(M = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P2),aa_nat_int(power_power_int(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_2933_power__minus__mult,axiom,
    ! [N: nat,A2: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N) ) ) ).

% power_minus_mult
tff(fact_2934_power__minus__mult,axiom,
    ! [N: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),N) ) ) ).

% power_minus_mult
tff(fact_2935_power__minus__mult,axiom,
    ! [N: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) ) ) ).

% power_minus_mult
tff(fact_2936_power__minus__mult,axiom,
    ! [N: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_rat(power_power_rat(A2),N) ) ) ).

% power_minus_mult
tff(fact_2937_power__minus__mult,axiom,
    ! [N: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N) ) ) ).

% power_minus_mult
tff(fact_2938_power__minus__mult,axiom,
    ! [N: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A2) = aa_nat_int(power_power_int(A2),N) ) ) ).

% power_minus_mult
tff(fact_2939_split__div_H,axiom,
    ! [P: fun_nat_o,M: nat,N: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N))
    <=> ( ( ( N = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [Q4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q4)),M)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,Q4)))
            & aa_nat_o(P,Q4) ) ) ) ).

% split_div'
tff(fact_2940_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,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv2),Uw2)),Xb) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_2941_power2__sum,axiom,
    ! [Xb: complex,Y: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_2942_power2__sum,axiom,
    ! [Xb: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_2943_power2__sum,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(plus_p361126936061061375l_num1(Xb,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = plus_p361126936061061375l_num1(plus_p361126936061061375l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_2944_power2__sum,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_2945_power2__sum,axiom,
    ! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(plus_plus_rat(Xb,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = plus_plus_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_2946_power2__sum,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_2947_power2__sum,axiom,
    ! [Xb: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_2948_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A2: $o,Uw2: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,vEBT_Leaf((A2),(Uw2))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
tff(fact_2949_zero__le__even__power_H,axiom,
    ! [A2: real,N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% zero_le_even_power'
tff(fact_2950_zero__le__even__power_H,axiom,
    ! [A2: code_integer,N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% zero_le_even_power'
tff(fact_2951_zero__le__even__power_H,axiom,
    ! [A2: rat,N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% zero_le_even_power'
tff(fact_2952_zero__le__even__power_H,axiom,
    ! [A2: int,N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% zero_le_even_power'
tff(fact_2953_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] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(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)
tff(fact_2954_nat__bit__induct,axiom,
    ! [P: fun_nat_o,N: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N3: nat] :
            ( aa_nat_o(P,N3)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)) ) )
       => ( ! [N3: nat] :
              ( aa_nat_o(P,N3)
             => aa_nat_o(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3))) )
         => aa_nat_o(P,N) ) ) ) ).

% nat_bit_induct
tff(fact_2955_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] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(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)
tff(fact_2956_axxdiv2,axiom,
    ! [Xb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Xb)),Xb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = Xb )
      & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),Xb)),Xb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = Xb ) ) ).

% axxdiv2
tff(fact_2957_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_2958_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D22)
       => ? [E: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E),D1)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_2959_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D1)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D22)
       => ? [E: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E),D1)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_2960_two__realpow__ge__one,axiom,
    ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N)) ).

% two_realpow_ge_one
tff(fact_2961_power2__diff,axiom,
    ! [Xb: complex,Y: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_2962_power2__diff,axiom,
    ! [Xb: code_integer,Y: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_2963_power2__diff,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(minus_4019991460397169231l_num1(Xb,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = minus_4019991460397169231l_num1(plus_p361126936061061375l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_2964_power2__diff,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_2965_power2__diff,axiom,
    ! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(minus_minus_rat(Xb,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = minus_minus_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_2966_power2__diff,axiom,
    ! [Xb: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_2967_odd__0__le__power__imp__0__le,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2968_odd__0__le__power__imp__0__le,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2969_odd__0__le__power__imp__0__le,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2970_odd__0__le__power__imp__0__le,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2971_odd__power__less__zero,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_z3403309356797280102nteger) ) ).

% odd_power_less_zero
tff(fact_2972_odd__power__less__zero,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_2973_odd__power__less__zero,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_2974_odd__power__less__zero,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_2975_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,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz)),Xb) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_2976_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] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)),Vj) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
tff(fact_2977_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] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)),Vi) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_2978_nat__div__eq__Suc__0__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% nat_div_eq_Suc_0_iff
tff(fact_2979_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,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)),Xb) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_2980_field__sum__of__halves,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(divide_divide_real(Xb),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(divide_divide_real(Xb),numeral_numeral_real(aa_num_num(bit0,one)))) = Xb ).

% field_sum_of_halves
tff(fact_2981_field__sum__of__halves,axiom,
    ! [Xb: rat] : plus_plus_rat(divide_divide_rat(Xb,numeral_numeral_rat(aa_num_num(bit0,one))),divide_divide_rat(Xb,numeral_numeral_rat(aa_num_num(bit0,one)))) = Xb ).

% field_sum_of_halves
tff(fact_2982_arith__geo__mean,axiom,
    ! [U: real,Xb: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(times_times_real(Xb),Y) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_2983_arith__geo__mean,axiom,
    ! [U: rat,Xb: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(Xb),Y) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),divide_divide_rat(plus_plus_rat(Xb,Y),numeral_numeral_rat(aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_2984_divmod__step__eq,axiom,
    ! [L: num,Q2: int,R: int] :
      aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,R),aa_num_int(numeral_numeral_int,L))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2)),R)) ).

% divmod_step_eq
tff(fact_2985_divmod__step__eq,axiom,
    ! [L: num,Q2: nat,R: nat] :
      aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(L),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Q2),R)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R),aa_num_nat(numeral_numeral_nat,L))),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q2)),R)) ).

% divmod_step_eq
tff(fact_2986_divmod__step__eq,axiom,
    ! [L: num,Q2: code_integer,R: code_integer] :
      unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q2,R)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q2)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(R),numera6620942414471956472nteger(L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q2),R)) ).

% divmod_step_eq
tff(fact_2987_power__2__mult__step__le,axiom,
    ! [N5: nat,N: nat,K4: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)),K4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),K))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N5)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K4),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),K)) ) ) ).

% power_2_mult_step_le
tff(fact_2988_nat__less__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% nat_less_power_trans
tff(fact_2989_sum__squares__bound,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),Xb)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_2990_sum__squares__bound,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),Xb)),Y)),plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_2991_nat__le__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% nat_le_power_trans
tff(fact_2992_nat__power__less__diff,axiom,
    ! [N: nat,Q2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ).

% nat_power_less_diff
tff(fact_2993_nat__add__offset__less,axiom,
    ! [Y: nat,N: nat,Xb: nat,M: nat,Sz: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
       => ( ( Sz = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),Y)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Sz)) ) ) ) ).

% nat_add_offset_less
tff(fact_2994_two__powr__height__bound__deg,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% two_powr_height_bound_deg
tff(fact_2995_height__compose__summary,axiom,
    ! [Summary2: vEBT_VEBT,Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Summary2))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary2))) ).

% height_compose_summary
tff(fact_2996_height__compose__child,axiom,
    ! [Tb: vEBT_VEBT,TreeList: list_VEBT_VEBT,Info2: option4927543243414619207at_nat,Deg2: nat,Summary2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Tb),set_VEBT_VEBT2(TreeList))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary2))) ) ).

% height_compose_child
tff(fact_2997_delete__bound__height_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Tb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb))) ) ).

% delete_bound_height'
tff(fact_2998_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% semiring_norm(13)
tff(fact_2999_semiring__norm_I11_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(times_times_num,M),one) = M ).

% semiring_norm(11)
tff(fact_3000_semiring__norm_I12_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,one),N) = N ).

% semiring_norm(12)
tff(fact_3001_num__double,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,one)),N) = aa_num_num(bit0,N) ).

% num_double
tff(fact_3002_power__mult__numeral,axiom,
    ! [A2: nat,M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% power_mult_numeral
tff(fact_3003_power__mult__numeral,axiom,
    ! [A2: int,M: num,N: num] : aa_nat_int(power_power_int(aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% power_mult_numeral
tff(fact_3004_power__mult__numeral,axiom,
    ! [A2: real,M: num,N: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% power_mult_numeral
tff(fact_3005_power__mult__numeral,axiom,
    ! [A2: complex,M: num,N: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% power_mult_numeral
tff(fact_3006_power__mult__numeral,axiom,
    ! [A2: code_integer,M: num,N: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% power_mult_numeral
tff(fact_3007_less__eq__real__def,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_real_def
tff(fact_3008_real__arch__pow__inv,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ? [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N3)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_3009_real__arch__pow,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ? [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N3)) ) ).

% real_arch_pow
tff(fact_3010_times__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(times_times_int,K),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_3011_times__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),L) = zero_zero_int ).

% times_int_code(2)
tff(fact_3012_imult__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( times_7803423173614009249d_enat(M,N) = zero_z5237406670263579293d_enat )
    <=> ( ( M = zero_z5237406670263579293d_enat )
        | ( N = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_3013_four__x__squared,axiom,
    ! [Xb: real] : aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% four_x_squared
tff(fact_3014_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C: real,B2: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(times_times_real(A2),C))),aa_real_real(times_times_real(B2),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_3015_div__mult2__numeral__eq,axiom,
    ! [A2: nat,K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,L)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_3016_div__mult2__numeral__eq,axiom,
    ! [A2: int,K: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_3017_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_3018_zdiv__mult__self,axiom,
    ! [M: int,A2: int,N: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,M),N))),M) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),M)),N) ) ) ).

% zdiv_mult_self
tff(fact_3019_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(M,N))
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),M)
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),N) ) ) ).

% enat_0_less_mult_iff
tff(fact_3020_realpow__pos__nth2,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ? [R3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),aa_nat_nat(suc,N)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_3021_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q5: int,R5: int,Q2: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R5)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q2)),R))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),R)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),R5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3022_unique__quotient__lemma,axiom,
    ! [B2: int,Q5: int,R5: int,Q2: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R5)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q2)),R))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R5),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3023_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q2: int,R: int,B3: int,Q5: int,R5: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q2)),R) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R5) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R5)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R5)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),B2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3024_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q2: int,R: int,B3: int,Q5: int,R5: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q2)),R) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R5) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R5))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R5),B3)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),B2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3025_q__pos__lemma,axiom,
    ! [B3: int,Q5: int,R5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B3),Q5)),R5))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R5),B3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_3026_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_3027_zdiv__zmult2__eq,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_3028_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q2)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B2)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_3029_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q2)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),R)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_3030_split__zdiv,axiom,
    ! [P: fun_int_o,N: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => aa_int_o(P,I4) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => aa_int_o(P,I4) ) ) ) ) ).

% split_zdiv
tff(fact_3031_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_jb(L)),Qr) ).

% divmod_step_int_def
tff(fact_3032_z1pdiv2,axiom,
    ! [B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = B2 ).

% z1pdiv2
tff(fact_3033_pred__bound__height_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Tb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb))) ) ).

% pred_bound_height'
tff(fact_3034_succ_H__bound__height,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Tb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb))) ) ).

% succ'_bound_height
tff(fact_3035_member__bound__height_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,Tb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb))) ) ).

% member_bound_height'
tff(fact_3036_realpow__pos__nth__unique,axiom,
    ! [N: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
       => ? [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X2)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),N) = A2 )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4)
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y4),N) = A2 ) )
               => ( Y4 = X2 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_3037_realpow__pos__nth,axiom,
    ! [N: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),N) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_3038_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2))) ) ).

% height_node
tff(fact_3039_diff__diff__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),N) ) ) ).

% diff_diff_less
tff(fact_3040_neg__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),one_one_int)),A2) ) ) ).

% neg_zdiv_mult_2
tff(fact_3041_pos__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_3042_pos2,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% pos2
tff(fact_3043_n__less__equal__power__2,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% n_less_equal_power_2
tff(fact_3044_msrevs_I1_J,axiom,
    ! [N: nat,K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)),M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),K) ) ) ).

% msrevs(1)
tff(fact_3045_nat__mult__power__less__eq,axiom,
    ! [B2: nat,A2: nat,N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% nat_mult_power_less_eq
tff(fact_3046_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(L),Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_jc(L)),Qr) ).

% divmod_step_nat_def
tff(fact_3047_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_jb(L)),Qr) ).

% divmod_step_def
tff(fact_3048_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(L),Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_jc(L)),Qr) ).

% divmod_step_def
tff(fact_3049_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_jd(L),Qr) ).

% divmod_step_def
tff(fact_3050_cnt__bound_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Tb)),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N)),one_one_real))) ) ).

% cnt_bound'
tff(fact_3051_real__average__minus__first,axiom,
    ! [A2: real,B2: real] : aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),numeral_numeral_real(aa_num_num(bit0,one)))),A2) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(B2),A2)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_3052_real__average__minus__second,axiom,
    ! [B2: real,A2: real] : aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2)),numeral_numeral_real(aa_num_num(bit0,one)))),A2) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(B2),A2)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_3053_bset_I6_J,axiom,
    ! [D4: int,B5: set_int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B5)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Tb)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)),Tb) ) ) ) ).

% bset(6)
tff(fact_3054_bset_I8_J,axiom,
    ! [D4: int,Tb: int,B5: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Tb),one_one_int)),B5)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B5)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Tb),X3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Tb),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) ) ) ) ).

% bset(8)
tff(fact_3055_aset_I6_J,axiom,
    ! [D4: int,Tb: int,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Tb),one_one_int)),A3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A3)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Tb)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)),Tb) ) ) ) ) ).

% aset(6)
tff(fact_3056_aset_I8_J,axiom,
    ! [D4: int,A3: set_int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A3)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Tb),X3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Tb),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) ) ) ).

% aset(8)
tff(fact_3057_pred__empty,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( aa_nat_option_nat(vEBT_vebt_pred(Tb),Xb) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_je(Tb),Xb)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_3058_cnt__non__neg,axiom,
    ! [Tb: vEBT_VEBT] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Tb)) ).

% cnt_non_neg
tff(fact_3059_mint__corr__help__empty,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( vEBT_vebt_mint(Tb) = none_nat )
       => ( vEBT_VEBT_set_vebt(Tb) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_3060_maxt__corr__help__empty,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( vEBT_vebt_maxt(Tb) = none_nat )
       => ( vEBT_VEBT_set_vebt(Tb) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_3061_succ__empty,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( aa_nat_option_nat(vEBT_vebt_succ(Tb),Xb) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_jf(Tb),Xb)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_3062_atLeastatMost__empty__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( set_or4548717258645045905et_nat(A2,B2) = bot_bot_set_set_nat )
    <=> ~ aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_3063_atLeastatMost__empty__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( set_or633870826150836451st_rat(A2,B2) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_3064_atLeastatMost__empty__iff,axiom,
    ! [A2: num,B2: num] :
      ( ( set_or7049704709247886629st_num(A2,B2) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_3065_atLeastatMost__empty__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A2),B2) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_3066_atLeastatMost__empty__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( set_or1266510415728281911st_int(A2,B2) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_3067_atLeastatMost__empty__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( set_or1222579329274155063t_real(A2,B2) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_3068_atLeastatMost__empty__iff2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A2,B2) )
    <=> ~ aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_3069_atLeastatMost__empty__iff2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A2,B2) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_3070_atLeastatMost__empty__iff2,axiom,
    ! [A2: num,B2: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A2,B2) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_3071_atLeastatMost__empty__iff2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A2),B2) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_3072_atLeastatMost__empty__iff2,axiom,
    ! [A2: int,B2: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(A2,B2) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_3073_atLeastatMost__empty__iff2,axiom,
    ! [A2: real,B2: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A2,B2) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_3074_atLeastatMost__empty,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => ( set_or633870826150836451st_rat(A2,B2) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_3075_atLeastatMost__empty,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2)
     => ( set_or7049704709247886629st_num(A2,B2) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_3076_atLeastatMost__empty,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,A2),B2) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_3077_atLeastatMost__empty,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
     => ( set_or1266510415728281911st_int(A2,B2) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_3078_atLeastatMost__empty,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => ( set_or1222579329274155063t_real(A2,B2) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_3079_not__real__square__gt__zero,axiom,
    ! [Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(Xb),Xb))
    <=> ( Xb = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_3080_minf_I7_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Tb),X3) ) ).

% minf(7)
tff(fact_3081_minf_I7_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Tb),X3) ) ).

% minf(7)
tff(fact_3082_minf_I7_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Tb),X3) ) ).

% minf(7)
tff(fact_3083_minf_I7_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Tb),X3) ) ).

% minf(7)
tff(fact_3084_minf_I7_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Tb),X3) ) ).

% minf(7)
tff(fact_3085_minf_I5_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Tb) ) ).

% minf(5)
tff(fact_3086_minf_I5_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Tb) ) ).

% minf(5)
tff(fact_3087_minf_I5_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Tb) ) ).

% minf(5)
tff(fact_3088_minf_I5_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Tb) ) ).

% minf(5)
tff(fact_3089_minf_I5_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Tb) ) ).

% minf(5)
tff(fact_3090_minf_I4_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(4)
tff(fact_3091_minf_I4_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(4)
tff(fact_3092_minf_I4_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(4)
tff(fact_3093_minf_I4_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(4)
tff(fact_3094_minf_I4_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(4)
tff(fact_3095_minf_I3_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(3)
tff(fact_3096_minf_I3_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(3)
tff(fact_3097_minf_I3_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(3)
tff(fact_3098_minf_I3_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(3)
tff(fact_3099_minf_I3_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
     => ( X3 != Tb ) ) ).

% minf(3)
tff(fact_3100_minf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z4: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z4)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P3,X2) ) )
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z4)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q6,X2) ) )
       => ? [Z2: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
           => ( ( aa_real_o(P,X3)
                | aa_real_o(Q,X3) )
            <=> ( aa_real_o(P3,X3)
                | aa_real_o(Q6,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_3101_minf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z4)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P3,X2) ) )
     => ( ? [Z4: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z4)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q6,X2) ) )
       => ? [Z2: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
           => ( ( aa_rat_o(P,X3)
                | aa_rat_o(Q,X3) )
            <=> ( aa_rat_o(P3,X3)
                | aa_rat_o(Q6,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_3102_minf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z4: num] :
        ! [X2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z4)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P3,X2) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z4)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q6,X2) ) )
       => ? [Z2: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
           => ( ( aa_num_o(P,X3)
                | aa_num_o(Q,X3) )
            <=> ( aa_num_o(P3,X3)
                | aa_num_o(Q6,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_3103_minf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z4)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P3,X2) ) )
     => ( ? [Z4: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z4)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q6,X2) ) )
       => ? [Z2: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
           => ( ( aa_nat_o(P,X3)
                | aa_nat_o(Q,X3) )
            <=> ( aa_nat_o(P3,X3)
                | aa_nat_o(Q6,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_3104_minf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z4: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P3,X2) ) )
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q6,X2) ) )
       => ? [Z2: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
           => ( ( aa_int_o(P,X3)
                | aa_int_o(Q,X3) )
            <=> ( aa_int_o(P3,X3)
                | aa_int_o(Q6,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_3105_minf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z4: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z4)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P3,X2) ) )
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z4)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q6,X2) ) )
       => ? [Z2: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
           => ( ( aa_real_o(P,X3)
                & aa_real_o(Q,X3) )
            <=> ( aa_real_o(P3,X3)
                & aa_real_o(Q6,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_3106_minf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z4)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P3,X2) ) )
     => ( ? [Z4: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z4)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q6,X2) ) )
       => ? [Z2: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
           => ( ( aa_rat_o(P,X3)
                & aa_rat_o(Q,X3) )
            <=> ( aa_rat_o(P3,X3)
                & aa_rat_o(Q6,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_3107_minf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z4: num] :
        ! [X2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z4)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P3,X2) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z4)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q6,X2) ) )
       => ? [Z2: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
           => ( ( aa_num_o(P,X3)
                & aa_num_o(Q,X3) )
            <=> ( aa_num_o(P3,X3)
                & aa_num_o(Q6,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_3108_minf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z4)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P3,X2) ) )
     => ( ? [Z4: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z4)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q6,X2) ) )
       => ? [Z2: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
           => ( ( aa_nat_o(P,X3)
                & aa_nat_o(Q,X3) )
            <=> ( aa_nat_o(P3,X3)
                & aa_nat_o(Q6,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_3109_minf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z4: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P3,X2) ) )
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q6,X2) ) )
       => ? [Z2: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
           => ( ( aa_int_o(P,X3)
                & aa_int_o(Q,X3) )
            <=> ( aa_int_o(P3,X3)
                & aa_int_o(Q6,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_3110_pinf_I7_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Tb),X3) ) ).

% pinf(7)
tff(fact_3111_pinf_I7_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Tb),X3) ) ).

% pinf(7)
tff(fact_3112_pinf_I7_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Tb),X3) ) ).

% pinf(7)
tff(fact_3113_pinf_I7_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Tb),X3) ) ).

% pinf(7)
tff(fact_3114_pinf_I7_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Tb),X3) ) ).

% pinf(7)
tff(fact_3115_pinf_I5_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Tb) ) ).

% pinf(5)
tff(fact_3116_pinf_I5_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Tb) ) ).

% pinf(5)
tff(fact_3117_pinf_I5_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Tb) ) ).

% pinf(5)
tff(fact_3118_pinf_I5_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Tb) ) ).

% pinf(5)
tff(fact_3119_pinf_I5_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Tb) ) ).

% pinf(5)
tff(fact_3120_pinf_I4_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(4)
tff(fact_3121_pinf_I4_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(4)
tff(fact_3122_pinf_I4_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(4)
tff(fact_3123_pinf_I4_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(4)
tff(fact_3124_pinf_I4_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(4)
tff(fact_3125_pinf_I3_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(3)
tff(fact_3126_pinf_I3_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(3)
tff(fact_3127_pinf_I3_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(3)
tff(fact_3128_pinf_I3_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(3)
tff(fact_3129_pinf_I3_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
     => ( X3 != Tb ) ) ).

% pinf(3)
tff(fact_3130_pinf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z4: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X2)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P3,X2) ) )
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X2)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q6,X2) ) )
       => ? [Z2: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
           => ( ( aa_real_o(P,X3)
                | aa_real_o(Q,X3) )
            <=> ( aa_real_o(P3,X3)
                | aa_real_o(Q6,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_3131_pinf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X2)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P3,X2) ) )
     => ( ? [Z4: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X2)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q6,X2) ) )
       => ? [Z2: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
           => ( ( aa_rat_o(P,X3)
                | aa_rat_o(Q,X3) )
            <=> ( aa_rat_o(P3,X3)
                | aa_rat_o(Q6,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_3132_pinf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z4: num] :
        ! [X2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X2)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P3,X2) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X2)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q6,X2) ) )
       => ? [Z2: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
           => ( ( aa_num_o(P,X3)
                | aa_num_o(Q,X3) )
            <=> ( aa_num_o(P3,X3)
                | aa_num_o(Q6,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_3133_pinf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X2)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P3,X2) ) )
     => ( ? [Z4: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X2)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q6,X2) ) )
       => ? [Z2: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
           => ( ( aa_nat_o(P,X3)
                | aa_nat_o(Q,X3) )
            <=> ( aa_nat_o(P3,X3)
                | aa_nat_o(Q6,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_3134_pinf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z4: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P3,X2) ) )
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q6,X2) ) )
       => ? [Z2: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
           => ( ( aa_int_o(P,X3)
                | aa_int_o(Q,X3) )
            <=> ( aa_int_o(P3,X3)
                | aa_int_o(Q6,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_3135_pinf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z4: real] :
        ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X2)
         => ( aa_real_o(P,X2)
          <=> aa_real_o(P3,X2) ) )
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X2)
           => ( aa_real_o(Q,X2)
            <=> aa_real_o(Q6,X2) ) )
       => ? [Z2: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
           => ( ( aa_real_o(P,X3)
                & aa_real_o(Q,X3) )
            <=> ( aa_real_o(P3,X3)
                & aa_real_o(Q6,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_3136_pinf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X2)
         => ( aa_rat_o(P,X2)
          <=> aa_rat_o(P3,X2) ) )
     => ( ? [Z4: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X2)
           => ( aa_rat_o(Q,X2)
            <=> aa_rat_o(Q6,X2) ) )
       => ? [Z2: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
           => ( ( aa_rat_o(P,X3)
                & aa_rat_o(Q,X3) )
            <=> ( aa_rat_o(P3,X3)
                & aa_rat_o(Q6,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_3137_pinf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z4: num] :
        ! [X2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X2)
         => ( aa_num_o(P,X2)
          <=> aa_num_o(P3,X2) ) )
     => ( ? [Z4: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X2)
           => ( aa_num_o(Q,X2)
            <=> aa_num_o(Q6,X2) ) )
       => ? [Z2: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
           => ( ( aa_num_o(P,X3)
                & aa_num_o(Q,X3) )
            <=> ( aa_num_o(P3,X3)
                & aa_num_o(Q6,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_3138_pinf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X2)
         => ( aa_nat_o(P,X2)
          <=> aa_nat_o(P3,X2) ) )
     => ( ? [Z4: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X2)
           => ( aa_nat_o(Q,X2)
            <=> aa_nat_o(Q6,X2) ) )
       => ? [Z2: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
           => ( ( aa_nat_o(P,X3)
                & aa_nat_o(Q,X3) )
            <=> ( aa_nat_o(P3,X3)
                & aa_nat_o(Q6,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_3139_pinf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z4: int] :
        ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
         => ( aa_int_o(P,X2)
          <=> aa_int_o(P3,X2) ) )
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
           => ( aa_int_o(Q,X2)
            <=> aa_int_o(Q6,X2) ) )
       => ? [Z2: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
           => ( ( aa_int_o(P,X3)
                & aa_int_o(Q,X3) )
            <=> ( aa_int_o(P3,X3)
                & aa_int_o(Q6,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_3140_mult__commute__abs,axiom,
    ! [C: real] : aTP_Lamm_jg(C) = times_times_real(C) ).

% mult_commute_abs
tff(fact_3141_mult__commute__abs,axiom,
    ! [C: rat] : aTP_Lamm_jh(C) = times_times_rat(C) ).

% mult_commute_abs
tff(fact_3142_mult__commute__abs,axiom,
    ! [C: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ji,C) = aa_nat_fun_nat_nat(times_times_nat,C) ).

% mult_commute_abs
tff(fact_3143_mult__commute__abs,axiom,
    ! [C: int] : aa_int_fun_int_int(aTP_Lamm_jj,C) = aa_int_fun_int_int(times_times_int,C) ).

% mult_commute_abs
tff(fact_3144_minf_I8_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Tb),X3) ) ).

% minf(8)
tff(fact_3145_minf_I8_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Tb),X3) ) ).

% minf(8)
tff(fact_3146_minf_I8_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Tb),X3) ) ).

% minf(8)
tff(fact_3147_minf_I8_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Tb),X3) ) ).

% minf(8)
tff(fact_3148_minf_I8_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Tb),X3) ) ).

% minf(8)
tff(fact_3149_minf_I6_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Tb) ) ).

% minf(6)
tff(fact_3150_minf_I6_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Tb) ) ).

% minf(6)
tff(fact_3151_minf_I6_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Tb) ) ).

% minf(6)
tff(fact_3152_minf_I6_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Tb) ) ).

% minf(6)
tff(fact_3153_minf_I6_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Tb) ) ).

% minf(6)
tff(fact_3154_pinf_I8_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Tb),X3) ) ).

% pinf(8)
tff(fact_3155_pinf_I8_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Tb),X3) ) ).

% pinf(8)
tff(fact_3156_pinf_I8_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Tb),X3) ) ).

% pinf(8)
tff(fact_3157_pinf_I8_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Tb),X3) ) ).

% pinf(8)
tff(fact_3158_pinf_I8_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Tb),X3) ) ).

% pinf(8)
tff(fact_3159_pinf_I6_J,axiom,
    ! [Tb: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Tb) ) ).

% pinf(6)
tff(fact_3160_pinf_I6_J,axiom,
    ! [Tb: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Tb) ) ).

% pinf(6)
tff(fact_3161_pinf_I6_J,axiom,
    ! [Tb: num] :
    ? [Z2: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Tb) ) ).

% pinf(6)
tff(fact_3162_pinf_I6_J,axiom,
    ! [Tb: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Tb) ) ).

% pinf(6)
tff(fact_3163_pinf_I6_J,axiom,
    ! [Tb: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Tb) ) ).

% pinf(6)
tff(fact_3164_imp__le__cong,axiom,
    ! [Xb: int,X4: int,P: $o,P3: $o] :
      ( ( Xb = X4 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
           => (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
           => (P3) ) ) ) ) ).

% imp_le_cong
tff(fact_3165_conj__le__cong,axiom,
    ! [Xb: int,X4: int,P: $o,P3: $o] :
      ( ( Xb = X4 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
            & (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
            & (P3) ) ) ) ) ).

% conj_le_cong
tff(fact_3166_plusinfinity,axiom,
    ! [D: int,P3: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(P3,X2)
          <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z4: int] :
            ! [X2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
             => ( aa_int_o(P,X2)
              <=> aa_int_o(P3,X2) ) )
         => ( ? [X_13: int] : aa_int_o(P3,X_13)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_3167_minusinfinity,axiom,
    ! [D: int,P1: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(P1,X2)
          <=> aa_int_o(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z4: int] :
            ! [X2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
             => ( aa_int_o(P,X2)
              <=> aa_int_o(P1,X2) ) )
         => ( ? [X_13: int] : aa_int_o(P1,X_13)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_3168_bset_I1_J,axiom,
    ! [D4: int,B5: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B5)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B5)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B5)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( ( aa_int_o(P,X3)
                & aa_int_o(Q,X3) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) ) ) ) ) ).

% bset(1)
tff(fact_3169_bset_I2_J,axiom,
    ! [D4: int,B5: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),B5)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),B5)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B5)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( ( aa_int_o(P,X3)
                | aa_int_o(Q,X3) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) ) ) ) ) ).

% bset(2)
tff(fact_3170_aset_I1_J,axiom,
    ! [D4: int,A3: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A3)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A3)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A3)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( ( aa_int_o(P,X3)
                & aa_int_o(Q,X3) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) ) ) ) ) ).

% aset(1)
tff(fact_3171_aset_I2_J,axiom,
    ! [D4: int,A3: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( aa_set_int_o(member_int(Xb3),A3)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa) ) ) )
         => ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( aa_set_int_o(member_int(Xb3),A3)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa) ) ) )
           => ( aa_int_o(Q,X2)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A3)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( ( aa_int_o(P,X3)
                | aa_int_o(Q,X3) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) ) ) ) ) ).

% aset(2)
tff(fact_3172_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int] :
            ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X3: int] :
              ( aa_int_o(P,X3)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3173_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int] :
            ( aa_int_o(P,X2)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X3: int] :
              ( aa_int_o(P,X3)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_3174_periodic__finite__ex,axiom,
    ! [D: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(P,X2)
          <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [X_12: int] : aa_int_o(P,X_12)
        <=> ? [X: int] :
              ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D))
              & aa_int_o(P,X) ) ) ) ) ).

% periodic_finite_ex
tff(fact_3175_aset_I7_J,axiom,
    ! [D4: int,A3: set_int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A3)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Tb),X3)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,Tb),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) ) ) ).

% aset(7)
tff(fact_3176_aset_I5_J,axiom,
    ! [D4: int,Tb: int,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(Tb),A3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A3)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Tb)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)),Tb) ) ) ) ) ).

% aset(5)
tff(fact_3177_aset_I4_J,axiom,
    ! [D4: int,Tb: int,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(Tb),A3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A3)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( ( X3 != Tb )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4) != Tb ) ) ) ) ) ).

% aset(4)
tff(fact_3178_aset_I3_J,axiom,
    ! [D4: int,Tb: int,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Tb),one_one_int)),A3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A3)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( ( X3 = Tb )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4) = Tb ) ) ) ) ) ).

% aset(3)
tff(fact_3179_bset_I7_J,axiom,
    ! [D4: int,Tb: int,B5: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(Tb),B5)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B5)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Tb),X3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,Tb),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) ) ) ) ).

% bset(7)
tff(fact_3180_bset_I5_J,axiom,
    ! [D4: int,B5: set_int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B5)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Tb)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)),Tb) ) ) ) ).

% bset(5)
tff(fact_3181_bset_I4_J,axiom,
    ! [D4: int,Tb: int,B5: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(Tb),B5)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B5)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( ( X3 != Tb )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4) != Tb ) ) ) ) ) ).

% bset(4)
tff(fact_3182_bset_I3_J,axiom,
    ! [D4: int,Tb: int,B5: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Tb),one_one_int)),B5)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B5)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( ( X3 = Tb )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4) = Tb ) ) ) ) ) ).

% bset(3)
tff(fact_3183_cppi,axiom,
    ! [D4: int,P: fun_int_o,P3: fun_int_o,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
           => ( aa_int_o(P,X2)
            <=> aa_int_o(P3,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4))
                 => ! [Xb3: int] :
                      ( aa_set_int_o(member_int(Xb3),A3)
                     => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa) ) ) )
             => ( aa_int_o(P,X2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D4)) ) )
         => ( ! [X2: int,K2: int] :
                ( aa_int_o(P3,X2)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X: int] :
                    ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D4))
                    & aa_int_o(P3,X) )
                | ? [X: int] :
                    ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D4))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),A3)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_3184_cpmi,axiom,
    ! [D4: int,P: fun_int_o,P3: fun_int_o,B5: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
           => ( aa_int_o(P,X2)
            <=> aa_int_o(P3,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D4))
                 => ! [Xb3: int] :
                      ( aa_set_int_o(member_int(Xb3),B5)
                     => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa) ) ) )
             => ( aa_int_o(P,X2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D4)) ) )
         => ( ! [X2: int,K2: int] :
                ( aa_int_o(P3,X2)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X: int] :
                    ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D4))
                    & aa_int_o(P3,X) )
                | ? [X: int] :
                    ( aa_set_int_o(member_int(X),set_or1266510415728281911st_int(one_one_int,D4))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),B5)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_3185_count__buildup,axiom,
    ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(N))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N))) ).

% count_buildup
tff(fact_3186_cnt__bound,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Tb)),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N)),aa_real_real(divide_divide_real(numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one)))))))) ) ).

% cnt_bound
tff(fact_3187_vebt__succ_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( aa_nat_option_nat(vEBT_vebt_succ(Xb),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = $ite((B),aa_nat_option_nat(some_nat,one_one_nat),none_nat) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B))),zero_zero_nat)) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( Xb = vEBT_Leaf((Uv),(Uw)) )
               => ! [N3: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N3) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw))),aa_nat_nat(suc,N3))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2) )
                 => ( ( Y = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2)),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( ( Y = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                aa_nat_option_nat(some_nat,Mi2),
                                $let(
                                  l: nat,
                                  l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                      $let(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_succ(nth_VEBT_VEBT(TreeList2,h)),l)),
                                          $let(
                                            sc: option_nat,
                                            sc:= aa_nat_option_nat(vEBT_vebt_succ(Summary),h),
                                            $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(sc))))) ) ) ),
                                      none_nat ) ) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_3188_psubsetI,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B5)
     => ( ( A3 != B5 )
       => aa_set_nat_o(ord_less_set_nat(A3),B5) ) ) ).

% psubsetI
tff(fact_3189_pos__mult__pos__ge,axiom,
    ! [Xb: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,N),one_one_int)),aa_int_int(aa_int_fun_int_int(times_times_int,N),Xb)) ) ) ).

% pos_mult_pos_ge
tff(fact_3190_vebt__pred_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( aa_nat_option_nat(vEBT_vebt_pred(Xb),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv))),zero_zero_nat)) ) ) )
         => ( ! [A: $o,Uw: $o] :
                ( ( Xb = vEBT_Leaf((A),(Uw)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = $ite((A),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(Uw))),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A: $o,B: $o] :
                  ( ( Xb = vEBT_Leaf((A),(B)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Y = $ite(
                              (B),
                              aa_nat_option_nat(some_nat,one_one_nat),
                              $ite((A),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2) )
                     => ( ( Y = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)),Xaa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Y = none_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                         => ( ( Y = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  aa_nat_option_nat(some_nat,Ma2),
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                          $ite(
                                            ( ( minlow != none_nat )
                                            & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,h))),aa_nat_option_nat(vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,h)),l)),
                                            $let(
                                              pr: option_nat,
                                              pr:= aa_nat_option_nat(vEBT_vebt_pred(Summary),h),
                                              $ite(
                                                pr = none_nat,
                                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Xaa),aa_nat_option_nat(some_nat,Mi2),none_nat),
                                                aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat(pr)))) ) ) ) ),
                                        none_nat ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_3191_set__bit__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] : bit_se4894374433684937756l_num1(zero_zero_nat,A2) = plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),divide1791077408188789448l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3192_set__bit__0,axiom,
    ! [A2: uint32] : bit_se6647067497041451410uint32(zero_zero_nat,A2) = plus_plus_uint32(one_one_uint32,aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3193_set__bit__0,axiom,
    ! [A2: int] : aa_int_int(bit_se7879613467334960850it_int(zero_zero_nat),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3194_set__bit__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_3195_buildup__nothing__in__min__max,axiom,
    ! [N: nat,Xb: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,vEBT_vebt_buildup(N)),Xb) ).

% buildup_nothing_in_min_max
tff(fact_3196_buildup__nothing__in__leaf,axiom,
    ! [N: nat,Xb: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,vEBT_vebt_buildup(N)),Xb) ).

% buildup_nothing_in_leaf
tff(fact_3197_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_3198_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,vEBT_vebt_buildup(N)),N) ) ).

% buildup_gives_valid
tff(fact_3199_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_num(bit1,M) = aa_num_num(bit1,N) )
    <=> ( M = N ) ) ).

% semiring_norm(90)
tff(fact_3200_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] : aa_num_num(bit0,M) != aa_num_num(bit1,N) ).

% semiring_norm(88)
tff(fact_3201_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] : aa_num_num(bit1,M) != aa_num_num(bit0,N) ).

% semiring_norm(89)
tff(fact_3202_semiring__norm_I84_J,axiom,
    ! [N: num] : one != aa_num_num(bit1,N) ).

% semiring_norm(84)
tff(fact_3203_semiring__norm_I86_J,axiom,
    ! [M: num] : aa_num_num(bit1,M) != one ).

% semiring_norm(86)
tff(fact_3204_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se7879613467334960850it_int(N),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_3205_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se7879613467334960850it_int(N),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% set_bit_negative_int_iff
tff(fact_3206_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit1,N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N) ) ).

% semiring_norm(80)
tff(fact_3207_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit1,N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N) ) ).

% semiring_norm(73)
tff(fact_3208_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% semiring_norm(9)
tff(fact_3209_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% semiring_norm(7)
tff(fact_3210_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,M)),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),aa_num_num(bit1,N))) ).

% semiring_norm(14)
tff(fact_3211_semiring__norm_I15_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),N)) ).

% semiring_norm(15)
tff(fact_3212_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit0,N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N) ) ).

% semiring_norm(81)
tff(fact_3213_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit1,N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N) ) ).

% semiring_norm(72)
tff(fact_3214_semiring__norm_I77_J,axiom,
    ! [N: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,N)) ).

% semiring_norm(77)
tff(fact_3215_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),one) ).

% semiring_norm(70)
tff(fact_3216_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_3217_semiring__norm_I3_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit0,N)) = aa_num_num(bit1,N) ).

% semiring_norm(3)
tff(fact_3218_semiring__norm_I4_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,N),one)) ).

% semiring_norm(4)
tff(fact_3219_semiring__norm_I5_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit0,M)),one) = aa_num_num(bit1,M) ).

% semiring_norm(5)
tff(fact_3220_semiring__norm_I8_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),one) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one)) ).

% semiring_norm(8)
tff(fact_3221_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)),one)) ).

% semiring_norm(10)
tff(fact_3222_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit1,M)),aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)),aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)))) ).

% semiring_norm(16)
tff(fact_3223_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit1,N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N) ) ).

% semiring_norm(79)
tff(fact_3224_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit0,N))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),N) ) ).

% semiring_norm(74)
tff(fact_3225_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_3226_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).

% div_Suc_eq_div_add3
tff(fact_3227_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_se7879613467334960850it_int(N),K)) ).

% set_bit_greater_eq
tff(fact_3228_xor__num_Ocases,axiom,
    ! [Xb: product_prod_num_num] :
      ( ( Xb != aa_num3156681643174609981um_num(product_Pair_num_num(one),one) )
     => ( ! [N3: num] : Xb != aa_num3156681643174609981um_num(product_Pair_num_num(one),aa_num_num(bit0,N3))
       => ( ! [N3: num] : Xb != aa_num3156681643174609981um_num(product_Pair_num_num(one),aa_num_num(bit1,N3))
         => ( ! [M3: num] : Xb != aa_num3156681643174609981um_num(product_Pair_num_num(aa_num_num(bit0,M3)),one)
           => ( ! [M3: num,N3: num] : Xb != aa_num3156681643174609981um_num(product_Pair_num_num(aa_num_num(bit0,M3)),aa_num_num(bit0,N3))
             => ( ! [M3: num,N3: num] : Xb != aa_num3156681643174609981um_num(product_Pair_num_num(aa_num_num(bit0,M3)),aa_num_num(bit1,N3))
               => ( ! [M3: num] : Xb != aa_num3156681643174609981um_num(product_Pair_num_num(aa_num_num(bit1,M3)),one)
                 => ( ! [M3: num,N3: num] : Xb != aa_num3156681643174609981um_num(product_Pair_num_num(aa_num_num(bit1,M3)),aa_num_num(bit0,N3))
                   => ~ ! [M3: num,N3: num] : Xb != aa_num3156681643174609981um_num(product_Pair_num_num(aa_num_num(bit1,M3)),aa_num_num(bit1,N3)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_3229_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] : Y != aa_num_num(bit0,X23)
       => ~ ! [X32: num] : Y != aa_num_num(bit1,X32) ) ) ).

% num.exhaust
tff(fact_3230_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_3231_numeral__Bit1,axiom,
    ! [N: num] : numera9087168376688890119uint32(aa_num_num(bit1,N)) = plus_plus_uint32(plus_plus_uint32(numera9087168376688890119uint32(N),numera9087168376688890119uint32(N)),one_one_uint32) ).

% numeral_Bit1
tff(fact_3232_numeral__Bit1,axiom,
    ! [N: num] : numera7442385471795722001l_num1(aa_num_num(bit1,N)) = plus_p361126936061061375l_num1(plus_p361126936061061375l_num1(numera7442385471795722001l_num1(N),numera7442385471795722001l_num1(N)),one_on7727431528512463931l_num1) ).

% numeral_Bit1
tff(fact_3233_numeral__Bit1,axiom,
    ! [N: num] : numeral_numeral_real(aa_num_num(bit1,N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(N)),numeral_numeral_real(N))),one_one_real) ).

% numeral_Bit1
tff(fact_3234_numeral__Bit1,axiom,
    ! [N: num] : numeral_numeral_rat(aa_num_num(bit1,N)) = plus_plus_rat(plus_plus_rat(numeral_numeral_rat(N),numeral_numeral_rat(N)),one_one_rat) ).

% numeral_Bit1
tff(fact_3235_numeral__Bit1,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,N))),one_one_nat) ).

% numeral_Bit1
tff(fact_3236_numeral__Bit1,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,N))),one_one_int) ).

% numeral_Bit1
tff(fact_3237_eval__nat__numeral_I3_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))) ).

% eval_nat_numeral(3)
tff(fact_3238_numeral__code_I3_J,axiom,
    ! [N: num] :
      numera9087168376688890119uint32(aa_num_num(bit1,N)) = $let(
        m: uint32,
        m:= numera9087168376688890119uint32(N),
        plus_plus_uint32(plus_plus_uint32(m,m),one_one_uint32) ) ).

% numeral_code(3)
tff(fact_3239_numeral__code_I3_J,axiom,
    ! [N: num] :
      numera7442385471795722001l_num1(aa_num_num(bit1,N)) = $let(
        m: word_N3645301735248828278l_num1,
        m:= numera7442385471795722001l_num1(N),
        plus_p361126936061061375l_num1(plus_p361126936061061375l_num1(m,m),one_on7727431528512463931l_num1) ) ).

% numeral_code(3)
tff(fact_3240_numeral__code_I3_J,axiom,
    ! [N: num] :
      numeral_numeral_real(aa_num_num(bit1,N)) = $let(
        m: real,
        m:= numeral_numeral_real(N),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m),m)),one_one_real) ) ).

% numeral_code(3)
tff(fact_3241_numeral__code_I3_J,axiom,
    ! [N: num] :
      numeral_numeral_rat(aa_num_num(bit1,N)) = $let(
        m: rat,
        m:= numeral_numeral_rat(N),
        plus_plus_rat(plus_plus_rat(m,m),one_one_rat) ) ).

% numeral_code(3)
tff(fact_3242_numeral__code_I3_J,axiom,
    ! [N: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N)) = $let(
        m: nat,
        m:= aa_num_nat(numeral_numeral_nat,N),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_3243_numeral__code_I3_J,axiom,
    ! [N: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,N),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m),m)),one_one_int) ) ).

% numeral_code(3)
tff(fact_3244_power__numeral__odd,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: complex,
        w:= aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_3245_power__numeral__odd,axiom,
    ! [Z: code_integer,W: num] :
      aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: code_integer,
        w:= aa_nat_Code_integer(power_8256067586552552935nteger(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_3246_power__numeral__odd,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(times_times_real(aa_real_real(times_times_real(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_3247_power__numeral__odd,axiom,
    ! [Z: rat,W: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_3248_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_3249_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: int,
        w:= aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_3250_psubset__imp__ex__mem,axiom,
    ! [A3: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A3),B5)
     => ? [B: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(B5,A3)) ) ).

% psubset_imp_ex_mem
tff(fact_3251_psubset__imp__ex__mem,axiom,
    ! [A3: set_int,B5: set_int] :
      ( aa_set_int_o(ord_less_set_int(A3),B5)
     => ? [B: int] : aa_set_int_o(member_int(B),minus_minus_set_int(B5,A3)) ) ).

% psubset_imp_ex_mem
tff(fact_3252_psubset__imp__ex__mem,axiom,
    ! [A3: set_real,B5: set_real] :
      ( aa_set_real_o(ord_less_set_real(A3),B5)
     => ? [B: real] : aa_set_real_o(member_real(B),minus_minus_set_real(B5,A3)) ) ).

% psubset_imp_ex_mem
tff(fact_3253_psubset__imp__ex__mem,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A3),B5)
     => ? [B: nat] : aa_set_nat_o(member_nat(B),minus_minus_set_nat(B5,A3)) ) ).

% psubset_imp_ex_mem
tff(fact_3254_psubsetD,axiom,
    ! [A3: set_nat,B5: set_nat,C: nat] :
      ( aa_set_nat_o(ord_less_set_nat(A3),B5)
     => ( aa_set_nat_o(member_nat(C),A3)
       => aa_set_nat_o(member_nat(C),B5) ) ) ).

% psubsetD
tff(fact_3255_psubsetD,axiom,
    ! [A3: set_VEBT_VEBT,B5: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A3),B5)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),A3)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C),B5) ) ) ).

% psubsetD
tff(fact_3256_psubsetD,axiom,
    ! [A3: set_int,B5: set_int,C: int] :
      ( aa_set_int_o(ord_less_set_int(A3),B5)
     => ( aa_set_int_o(member_int(C),A3)
       => aa_set_int_o(member_int(C),B5) ) ) ).

% psubsetD
tff(fact_3257_psubsetD,axiom,
    ! [A3: set_real,B5: set_real,C: real] :
      ( aa_set_real_o(ord_less_set_real(A3),B5)
     => ( aa_set_real_o(member_real(C),A3)
       => aa_set_real_o(member_real(C),B5) ) ) ).

% psubsetD
tff(fact_3258_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa_nat_nat(suc,zero_zero_nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(2)
tff(fact_3259_numeral__Bit1__div__2,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,N) ).

% numeral_Bit1_div_2
tff(fact_3260_numeral__Bit1__div__2,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,N) ).

% numeral_Bit1_div_2
tff(fact_3261_power3__eq__cube,axiom,
    ! [A2: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),A2)),A2) ).

% power3_eq_cube
tff(fact_3262_power3__eq__cube,axiom,
    ! [A2: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),A2)),A2) ).

% power3_eq_cube
tff(fact_3263_power3__eq__cube,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_real_real(times_times_real(aa_real_real(times_times_real(A2),A2)),A2) ).

% power3_eq_cube
tff(fact_3264_power3__eq__cube,axiom,
    ! [A2: rat] : aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A2),A2)),A2) ).

% power3_eq_cube
tff(fact_3265_power3__eq__cube,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),A2)),A2) ).

% power3_eq_cube
tff(fact_3266_power3__eq__cube,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),A2)),A2) ).

% power3_eq_cube
tff(fact_3267_numeral__3__eq__3,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))) ).

% numeral_3_eq_3
tff(fact_3268_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N) ).

% Suc3_eq_add_3
tff(fact_3269_Set_Oempty__def,axiom,
    bot_bot_set_complex = collect_complex(aTP_Lamm_jk) ).

% Set.empty_def
tff(fact_3270_Set_Oempty__def,axiom,
    bot_bo1796632182523588997nt_int = collec213857154873943460nt_int(aTP_Lamm_jl) ).

% Set.empty_def
tff(fact_3271_Set_Oempty__def,axiom,
    bot_bot_set_real = collect_real(aTP_Lamm_jm) ).

% Set.empty_def
tff(fact_3272_Set_Oempty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_fs) ).

% Set.empty_def
tff(fact_3273_Set_Oempty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_jn) ).

% Set.empty_def
tff(fact_3274_num_Osize_I6_J,axiom,
    ! [X33: num] : aa_num_nat(size_size_num,aa_num_num(bit1,X33)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X33)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_3275_less__set__def,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A3),B5)
    <=> ord_less_fun_nat_o(aTP_Lamm_ce(A3),aTP_Lamm_ce(B5)) ) ).

% less_set_def
tff(fact_3276_less__set__def,axiom,
    ! [A3: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A3),B5)
    <=> ord_le3534172602983796987VEBT_o(aTP_Lamm_cc(A3),aTP_Lamm_cc(B5)) ) ).

% less_set_def
tff(fact_3277_less__set__def,axiom,
    ! [A3: set_int,B5: set_int] :
      ( aa_set_int_o(ord_less_set_int(A3),B5)
    <=> ord_less_fun_int_o(aTP_Lamm_cf(A3),aTP_Lamm_cf(B5)) ) ).

% less_set_def
tff(fact_3278_less__set__def,axiom,
    ! [A3: set_real,B5: set_real] :
      ( aa_set_real_o(ord_less_set_real(A3),B5)
    <=> ord_less_fun_real_o(aTP_Lamm_cd(A3),aTP_Lamm_cd(B5)) ) ).

% less_set_def
tff(fact_3279_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),N) ).

% Suc_div_eq_add3_div
tff(fact_3280_not__psubset__empty,axiom,
    ! [A3: set_real] : ~ aa_set_real_o(ord_less_set_real(A3),bot_bot_set_real) ).

% not_psubset_empty
tff(fact_3281_not__psubset__empty,axiom,
    ! [A3: set_nat] : ~ aa_set_nat_o(ord_less_set_nat(A3),bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_3282_not__psubset__empty,axiom,
    ! [A3: set_int] : ~ aa_set_int_o(ord_less_set_int(A3),bot_bot_set_int) ).

% not_psubset_empty
tff(fact_3283_subset__iff__psubset__eq,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B5)
    <=> ( aa_set_nat_o(ord_less_set_nat(A3),B5)
        | ( A3 = B5 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_3284_subset__psubset__trans,axiom,
    ! [A3: set_nat,B5: set_nat,C4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B5)
     => ( aa_set_nat_o(ord_less_set_nat(B5),C4)
       => aa_set_nat_o(ord_less_set_nat(A3),C4) ) ) ).

% subset_psubset_trans
tff(fact_3285_subset__not__subset__eq,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A3),B5)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A3),B5)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(B5),A3) ) ) ).

% subset_not_subset_eq
tff(fact_3286_psubset__subset__trans,axiom,
    ! [A3: set_nat,B5: set_nat,C4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A3),B5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B5),C4)
       => aa_set_nat_o(ord_less_set_nat(A3),C4) ) ) ).

% psubset_subset_trans
tff(fact_3287_psubset__imp__subset,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A3),B5)
     => aa_set_nat_o(ord_less_eq_set_nat(A3),B5) ) ).

% psubset_imp_subset
tff(fact_3288_psubset__eq,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A3),B5)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A3),B5)
        & ( A3 != B5 ) ) ) ).

% psubset_eq
tff(fact_3289_psubsetE,axiom,
    ! [A3: set_nat,B5: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A3),B5)
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A3),B5)
         => aa_set_nat_o(ord_less_eq_set_nat(B5),A3) ) ) ).

% psubsetE
tff(fact_3290_small__powers__of__2,axiom,
    ! [Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),one_one_nat))) ) ).

% small_powers_of_2
tff(fact_3291_p2__eq__1,axiom,
    ! [N: nat] :
      ( ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N) = one_on7727431528512463931l_num1 )
    <=> ( N = zero_zero_nat ) ) ).

% p2_eq_1
tff(fact_3292_word__less__two__pow__divD,axiom,
    ! [Xb: word_N3645301735248828278l_num1,N: nat,M: nat] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),divide1791077408188789448l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
        & aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ) ) ).

% word_less_two_pow_divD
tff(fact_3293_less__1__helper,axiom,
    ! [N: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,N),M)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),one_one_int)),M) ) ).

% less_1_helper
tff(fact_3294_space__bound,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,Tb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),U)) ) ) ).

% space_bound
tff(fact_3295_space_H__bound,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Tb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),U)) ) ) ).

% space'_bound
tff(fact_3296_delete__bound__height,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Tb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))) ) ).

% delete_bound_height
tff(fact_3297_tdeletemimi,axiom,
    ! [Deg2: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Mi)),Deg2,TreeList),Summary2)),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% tdeletemimi
tff(fact_3298_vebt__buildup__bound,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),U)) ) ).

% vebt_buildup_bound
tff(fact_3299_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Xb),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B))),zero_zero_nat)) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( Xb = vEBT_Leaf((Uv),(Uw)) )
               => ! [N3: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N3) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw))),aa_nat_nat(suc,N3))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(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,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2)),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( ( Y = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                one_one_nat,
                                $let(
                                  l: nat,
                                  l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                      $let(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,nth_VEBT_VEBT(TreeList2,h)),l)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Summary),h)),one_one_nat) ) ),
                                      one_one_nat ) ) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_3300_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Xb),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv))),zero_zero_nat)) ) ) )
         => ( ! [A: $o,Uw: $o] :
                ( ( Xb = vEBT_Leaf((A),(Uw)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(Uw))),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A: $o,B: $o] :
                  ( ( Xb = vEBT_Leaf((A),(B)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(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,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)),Xaa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                         => ( ( Y = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  one_one_nat,
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                          $ite(
                                            ( ( minlow != none_nat )
                                            & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,nth_VEBT_VEBT(TreeList2,h)),l)),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Summary),h)),one_one_nat) ) ),
                                        one_one_nat ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_3301_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,Xb),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Y = one_one_nat )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                Xaa = Mi2,
                                zero_zero_nat,
                                $ite(
                                  Xaa = Ma2,
                                  zero_zero_nat,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                    zero_zero_nat,
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                      zero_zero_nat,
                                      $ite(
                                        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Xaa)
                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Ma2) ),
                                        $let(
                                          h: nat,
                                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r2,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_nat) ),
                                        zero_zero_nat ) ) ) ) )) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_3302_space__space_H,axiom,
    ! [Tb: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,Tb)),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Tb)) ).

% space_space'
tff(fact_3303_word__coorder_Oextremum__unique,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(A2),zero_z3563351764282998399l_num1)
    <=> ( A2 = zero_z3563351764282998399l_num1 ) ) ).

% word_coorder.extremum_unique
tff(fact_3304_word__le__0__iff,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),zero_z3563351764282998399l_num1)
    <=> ( Xb = zero_z3563351764282998399l_num1 ) ) ).

% word_le_0_iff
tff(fact_3305_div__of__0__id,axiom,
    ! [N: word_N3645301735248828278l_num1] : divide1791077408188789448l_num1(zero_z3563351764282998399l_num1,N) = zero_z3563351764282998399l_num1 ).

% div_of_0_id
tff(fact_3306_word__gt__0__no,axiom,
    ! [Y: num] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),numera7442385471795722001l_num1(Y))
    <=> ( zero_z3563351764282998399l_num1 != numera7442385471795722001l_num1(Y) ) ) ).

% word_gt_0_no
tff(fact_3307_word__less__1,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),one_on7727431528512463931l_num1)
    <=> ( Xb = zero_z3563351764282998399l_num1 ) ) ).

% word_less_1
tff(fact_3308_word__le__sub1__numberof,axiom,
    ! [W: num] :
      ( ( numera7442385471795722001l_num1(W) != zero_z3563351764282998399l_num1 )
     => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(one_on7727431528512463931l_num1),numera7442385471795722001l_num1(W))
      <=> aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(zero_z3563351764282998399l_num1),minus_4019991460397169231l_num1(numera7442385471795722001l_num1(W),one_on7727431528512463931l_num1)) ) ) ).

% word_le_sub1_numberof
tff(fact_3309_word__less__sub1__numberof,axiom,
    ! [W: num] :
      ( ( numera7442385471795722001l_num1(W) != zero_z3563351764282998399l_num1 )
     => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(one_on7727431528512463931l_num1),numera7442385471795722001l_num1(W))
      <=> aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),minus_4019991460397169231l_num1(numera7442385471795722001l_num1(W),one_on7727431528512463931l_num1)) ) ) ).

% word_less_sub1_numberof
tff(fact_3310_sub__wrap,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Z: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),minus_4019991460397169231l_num1(Xb,Z))
    <=> ( ( Z = zero_z3563351764282998399l_num1 )
        | aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),Z) ) ) ).

% sub_wrap
tff(fact_3311_less__1__simp,axiom,
    ! [N: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(minus_4019991460397169231l_num1(N,one_on7727431528512463931l_num1)),M)
    <=> ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(N),M)
        & ( N != zero_z3563351764282998399l_num1 ) ) ) ).

% less_1_simp
tff(fact_3312_le__m1__iff__lt,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),Xb)
    <=> ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Y),minus_4019991460397169231l_num1(Xb,one_on7727431528512463931l_num1))
      <=> aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Y),Xb) ) ) ).

% le_m1_iff_lt
tff(fact_3313_word__div__sub,axiom,
    ! [Y: word_N3645301735248828278l_num1,Xb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Y),Xb)
     => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),Y)
       => ( divide1791077408188789448l_num1(minus_4019991460397169231l_num1(Xb,Y),Y) = minus_4019991460397169231l_num1(divide1791077408188789448l_num1(Xb,Y),one_on7727431528512463931l_num1) ) ) ) ).

% word_div_sub
tff(fact_3314_word__less__nowrapI,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Z: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),minus_4019991460397169231l_num1(Z,K))
     => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(K),Z)
       => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),K)
         => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),plus_p361126936061061375l_num1(Xb,K)) ) ) ) ).

% word_less_nowrapI
tff(fact_3315_word__diff__less,axiom,
    ! [N: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),N)
     => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),M)
       => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(N),M)
         => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(minus_4019991460397169231l_num1(M,N)),M) ) ) ) ).

% word_diff_less
tff(fact_3316_plus__minus__not__NULL,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Ab: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),minus_4019991460397169231l_num1(Ab,C))
     => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(C),Ab)
       => ( ( C != zero_z3563351764282998399l_num1 )
         => ( plus_p361126936061061375l_num1(Xb,C) != zero_z3563351764282998399l_num1 ) ) ) ) ).

% plus_minus_not_NULL
tff(fact_3317_word__subset__less,axiom,
    ! [Xb: word_N3645301735248828278l_num1,R: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1,S: word_N3645301735248828278l_num1] :
      ( ord_le5203802739334966412l_num1(set_or6221694504095523457l_num1(Xb,minus_4019991460397169231l_num1(plus_p361126936061061375l_num1(Xb,R),one_on7727431528512463931l_num1)),set_or6221694504095523457l_num1(Y,minus_4019991460397169231l_num1(plus_p361126936061061375l_num1(Y,S),one_on7727431528512463931l_num1)))
     => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),minus_4019991460397169231l_num1(plus_p361126936061061375l_num1(Xb,R),one_on7727431528512463931l_num1))
       => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Y),minus_4019991460397169231l_num1(plus_p361126936061061375l_num1(Y,S),one_on7727431528512463931l_num1))
         => ( ( S != zero_z3563351764282998399l_num1 )
           => aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(R),S) ) ) ) ) ).

% word_subset_less
tff(fact_3318_plus__minus__not__NULL__ab,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Ab: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),minus_4019991460397169231l_num1(Ab,C))
     => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(C),Ab)
       => ( ( C != zero_z3563351764282998399l_num1 )
         => ( plus_p361126936061061375l_num1(Xb,C) != zero_z3563351764282998399l_num1 ) ) ) ) ).

% plus_minus_not_NULL_ab
tff(fact_3319_word__less__nowrapI_H,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Z: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),minus_4019991460397169231l_num1(Z,K))
     => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(K),Z)
       => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),K)
         => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),plus_p361126936061061375l_num1(Xb,K)) ) ) ) ).

% word_less_nowrapI'
tff(fact_3320_word__leq__le__minus__one,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),Y)
     => ( ( Xb != zero_z3563351764282998399l_num1 )
       => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(minus_4019991460397169231l_num1(Xb,one_on7727431528512463931l_num1)),Y) ) ) ).

% word_leq_le_minus_one
tff(fact_3321_word__leq__minus__one__le,axiom,
    ! [Y: word_N3645301735248828278l_num1,Xb: word_N3645301735248828278l_num1] :
      ( ( Y != zero_z3563351764282998399l_num1 )
     => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),minus_4019991460397169231l_num1(Y,one_on7727431528512463931l_num1))
       => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),Y) ) ) ).

% word_leq_minus_one_le
tff(fact_3322_word__sub__plus__one__nonzero,axiom,
    ! [N5: word_N3645301735248828278l_num1,N: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(N5),N)
     => ( ( N5 != zero_z3563351764282998399l_num1 )
       => ( plus_p361126936061061375l_num1(minus_4019991460397169231l_num1(N,N5),one_on7727431528512463931l_num1) != zero_z3563351764282998399l_num1 ) ) ) ).

% word_sub_plus_one_nonzero
tff(fact_3323_word__induct__less,axiom,
    ! [P: fun_wo7859709073469921915num1_o,M: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(P,zero_z3563351764282998399l_num1)
     => ( ! [N3: word_N3645301735248828278l_num1] :
            ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(N3),M)
           => ( aa_wor4452366487969921764num1_o(P,N3)
             => aa_wor4452366487969921764num1_o(P,plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,N3)) ) )
       => aa_wor4452366487969921764num1_o(P,M) ) ) ).

% word_induct_less
tff(fact_3324_word__overflow,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),plus_p361126936061061375l_num1(Xb,one_on7727431528512463931l_num1))
      | ( plus_p361126936061061375l_num1(Xb,one_on7727431528512463931l_num1) = zero_z3563351764282998399l_num1 ) ) ).

% word_overflow
tff(fact_3325_word__gr0__conv__Suc,axiom,
    ! [M: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),M)
     => ? [N3: word_N3645301735248828278l_num1] : M = plus_p361126936061061375l_num1(N3,one_on7727431528512463931l_num1) ) ).

% word_gr0_conv_Suc
tff(fact_3326_less__is__non__zero__p1,axiom,
    ! [A2: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(A2),K)
     => ( plus_p361126936061061375l_num1(A2,one_on7727431528512463931l_num1) != zero_z3563351764282998399l_num1 ) ) ).

% less_is_non_zero_p1
tff(fact_3327_word__induct2,axiom,
    ! [P: fun_wo7859709073469921915num1_o,N: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(P,zero_z3563351764282998399l_num1)
     => ( ! [N3: word_N3645301735248828278l_num1] :
            ( ( plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,N3) != zero_z3563351764282998399l_num1 )
           => ( aa_wor4452366487969921764num1_o(P,N3)
             => aa_wor4452366487969921764num1_o(P,plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,N3)) ) )
       => aa_wor4452366487969921764num1_o(P,N) ) ) ).

% word_induct2
tff(fact_3328_word__induct,axiom,
    ! [P: fun_wo7859709073469921915num1_o,M: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(P,zero_z3563351764282998399l_num1)
     => ( ! [N3: word_N3645301735248828278l_num1] :
            ( aa_wor4452366487969921764num1_o(P,N3)
           => aa_wor4452366487969921764num1_o(P,plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,N3)) )
       => aa_wor4452366487969921764num1_o(P,M) ) ) ).

% word_induct
tff(fact_3329_word__plus__one__nonzero,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),plus_p361126936061061375l_num1(Xb,Y))
     => ( ( Y != zero_z3563351764282998399l_num1 )
       => ( plus_p361126936061061375l_num1(Xb,one_on7727431528512463931l_num1) != zero_z3563351764282998399l_num1 ) ) ) ).

% word_plus_one_nonzero
tff(fact_3330_plus__one__helper2,axiom,
    ! [Xb: word_N3645301735248828278l_num1,N: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),N)
     => ( ( plus_p361126936061061375l_num1(N,one_on7727431528512463931l_num1) != zero_z3563351764282998399l_num1 )
       => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),plus_p361126936061061375l_num1(N,one_on7727431528512463931l_num1)) ) ) ).

% plus_one_helper2
tff(fact_3331_neq__0__no__wrap,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),plus_p361126936061061375l_num1(Xb,Y))
     => ( ( Xb != zero_z3563351764282998399l_num1 )
       => ( plus_p361126936061061375l_num1(Xb,Y) != zero_z3563351764282998399l_num1 ) ) ) ).

% neq_0_no_wrap
tff(fact_3332_div__le__mult,axiom,
    ! [I: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1,Xb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(I),divide1791077408188789448l_num1(K,Xb))
     => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),Xb)
       => aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(I),Xb)),K) ) ) ).

% div_le_mult
tff(fact_3333_div__lt__mult,axiom,
    ! [I: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1,Xb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(I),divide1791077408188789448l_num1(K,Xb))
     => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),Xb)
       => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(I),Xb)),K) ) ) ).

% div_lt_mult
tff(fact_3334_More__Word_Oword__div__mult,axiom,
    ! [C: word_N3645301735248828278l_num1,A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),C)
     => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(A2),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B2),C))
       => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(divide1791077408188789448l_num1(A2,C)),B2) ) ) ).

% More_Word.word_div_mult
tff(fact_3335_div__less__dividend__word,axiom,
    ! [Xb: word_N3645301735248828278l_num1,N: word_N3645301735248828278l_num1] :
      ( ( Xb != zero_z3563351764282998399l_num1 )
     => ( ( N != one_on7727431528512463931l_num1 )
       => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(divide1791077408188789448l_num1(Xb,N)),Xb) ) ) ).

% div_less_dividend_word
tff(fact_3336_gt0__iff__gem1,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),Xb)
    <=> aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(minus_4019991460397169231l_num1(Xb,one_on7727431528512463931l_num1)),Xb) ) ).

% gt0_iff_gem1
tff(fact_3337_word__less__sub1,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( ( Xb != zero_z3563351764282998399l_num1 )
     => ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(one_on7727431528512463931l_num1),Xb)
      <=> aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),minus_4019991460397169231l_num1(Xb,one_on7727431528512463931l_num1)) ) ) ).

% word_less_sub1
tff(fact_3338_div__word__self,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( W != zero_z3563351764282998399l_num1 )
     => ( divide1791077408188789448l_num1(W,W) = one_on7727431528512463931l_num1 ) ) ).

% div_word_self
tff(fact_3339_word__coorder_Oextremum,axiom,
    ! [A2: word_N3645301735248828278l_num1] : aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(zero_z3563351764282998399l_num1),A2) ).

% word_coorder.extremum
tff(fact_3340_word__coorder_Oextremum__uniqueI,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(A2),zero_z3563351764282998399l_num1)
     => ( A2 = zero_z3563351764282998399l_num1 ) ) ).

% word_coorder.extremum_uniqueI
tff(fact_3341_word__zero__le,axiom,
    ! [Y: word_N3645301735248828278l_num1] : aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(zero_z3563351764282998399l_num1),Y) ).

% word_zero_le
tff(fact_3342_lt1__neq0,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(one_on7727431528512463931l_num1),Xb)
    <=> ( Xb != zero_z3563351764282998399l_num1 ) ) ).

% lt1_neq0
tff(fact_3343_word__must__wrap,axiom,
    ! [Xb: word_N3645301735248828278l_num1,N: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(Xb),minus_4019991460397169231l_num1(N,one_on7727431528512463931l_num1))
     => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(N),Xb)
       => ( N = zero_z3563351764282998399l_num1 ) ) ) ).

% word_must_wrap
tff(fact_3344_word__sub__1__le,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( ( Xb != zero_z3563351764282998399l_num1 )
     => aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(minus_4019991460397169231l_num1(Xb,one_on7727431528512463931l_num1)),Xb) ) ).

% word_sub_1_le
tff(fact_3345_word__le__sub1,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( ( Xb != zero_z3563351764282998399l_num1 )
     => ( aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(one_on7727431528512463931l_num1),Xb)
      <=> aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(zero_z3563351764282998399l_num1),minus_4019991460397169231l_num1(Xb,one_on7727431528512463931l_num1)) ) ) ).

% word_le_sub1
tff(fact_3346_div__by__0__word,axiom,
    ! [Xb: word_N3645301735248828278l_num1] : divide1791077408188789448l_num1(Xb,zero_z3563351764282998399l_num1) = zero_z3563351764282998399l_num1 ).

% div_by_0_word
tff(fact_3347_set__diff__eq,axiom,
    ! [A3: set_VEBT_VEBT,B5: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A3,B5) = collect_VEBT_VEBT(aa_set2497645856053385599VEBT_o(aTP_Lamm_jo(A3),B5)) ).

% set_diff_eq
tff(fact_3348_set__diff__eq,axiom,
    ! [A3: set_real,B5: set_real] : minus_minus_set_real(A3,B5) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_jp(A3),B5)) ).

% set_diff_eq
tff(fact_3349_set__diff__eq,axiom,
    ! [A3: set_int,B5: set_int] : minus_minus_set_int(A3,B5) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_jq(A3),B5)) ).

% set_diff_eq
tff(fact_3350_set__diff__eq,axiom,
    ! [A3: set_complex,B5: set_complex] : minus_811609699411566653omplex(A3,B5) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_jr(A3),B5)) ).

% set_diff_eq
tff(fact_3351_set__diff__eq,axiom,
    ! [A3: set_Pr958786334691620121nt_int,B5: set_Pr958786334691620121nt_int] : minus_1052850069191792384nt_int(A3,B5) = collec213857154873943460nt_int(aa_set2224860389584585407_int_o(aTP_Lamm_js(A3),B5)) ).

% set_diff_eq
tff(fact_3352_set__diff__eq,axiom,
    ! [A3: set_nat,B5: set_nat] : minus_minus_set_nat(A3,B5) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_jt(A3),B5)) ).

% set_diff_eq
tff(fact_3353_minus__set__def,axiom,
    ! [A3: set_VEBT_VEBT,B5: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A3,B5) = collect_VEBT_VEBT(minus_4582106842932979072VEBT_o(aTP_Lamm_cc(A3),aTP_Lamm_cc(B5))) ).

% minus_set_def
tff(fact_3354_minus__set__def,axiom,
    ! [A3: set_real,B5: set_real] : minus_minus_set_real(A3,B5) = collect_real(minus_404528632214381484real_o(aTP_Lamm_cd(A3),aTP_Lamm_cd(B5))) ).

% minus_set_def
tff(fact_3355_minus__set__def,axiom,
    ! [A3: set_int,B5: set_int] : minus_minus_set_int(A3,B5) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_cf(A3),aTP_Lamm_cf(B5))) ).

% minus_set_def
tff(fact_3356_minus__set__def,axiom,
    ! [A3: set_complex,B5: set_complex] : minus_811609699411566653omplex(A3,B5) = collect_complex(minus_8169481405280535082plex_o(aTP_Lamm_cg(A3),aTP_Lamm_cg(B5))) ).

% minus_set_def
tff(fact_3357_minus__set__def,axiom,
    ! [A3: set_Pr958786334691620121nt_int,B5: set_Pr958786334691620121nt_int] : minus_1052850069191792384nt_int(A3,B5) = collec213857154873943460nt_int(minus_827889751674334651_int_o(aTP_Lamm_ch(A3),aTP_Lamm_ch(B5))) ).

% minus_set_def
tff(fact_3358_minus__set__def,axiom,
    ! [A3: set_nat,B5: set_nat] : minus_minus_set_nat(A3,B5) = collect_nat(minus_8972599623528345224_nat_o(aTP_Lamm_ce(A3),aTP_Lamm_ce(B5))) ).

% minus_set_def
tff(fact_3359_word__greater__zero__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),A2)
    <=> ( A2 != zero_z3563351764282998399l_num1 ) ) ).

% word_greater_zero_iff
tff(fact_3360_word__div__lt__eq__0,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),Y)
     => ( divide1791077408188789448l_num1(Xb,Y) = zero_z3563351764282998399l_num1 ) ) ).

% word_div_lt_eq_0
tff(fact_3361_word__gt__a__gt__0,axiom,
    ! [A2: word_N3645301735248828278l_num1,N: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(A2),N)
     => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),N) ) ).

% word_gt_a_gt_0
tff(fact_3362_word__less__div,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1(Xb,Y) = zero_z3563351764282998399l_num1 )
     => ( ( Y = zero_z3563351764282998399l_num1 )
        | aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),Y) ) ) ).

% word_less_div
tff(fact_3363_word__div__less,axiom,
    ! [W: word_N3645301735248828278l_num1,V: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(W),V)
     => ( divide1791077408188789448l_num1(W,V) = zero_z3563351764282998399l_num1 ) ) ).

% word_div_less
tff(fact_3364_word__neq__0__conv,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( W != zero_z3563351764282998399l_num1 )
    <=> aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),W) ) ).

% word_neq_0_conv
tff(fact_3365_word__gt__0,axiom,
    ! [Y: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),Y)
    <=> ( zero_z3563351764282998399l_num1 != Y ) ) ).

% word_gt_0
tff(fact_3366_word__coorder_Oextremum__strict,axiom,
    ! [A2: word_N3645301735248828278l_num1] : ~ aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(A2),zero_z3563351764282998399l_num1) ).

% word_coorder.extremum_strict
tff(fact_3367_word__not__simps_I1_J,axiom,
    ! [Xb: word_N3645301735248828278l_num1] : ~ aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xb),zero_z3563351764282998399l_num1) ).

% word_not_simps(1)
tff(fact_3368_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,
    ! [A2: $o,B2: $o,N: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,vEBT_Leaf((A2),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,N))) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
tff(fact_3369_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,
    ! [A2: $o,B2: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,vEBT_Leaf((A2),(B2))),zero_zero_nat) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
tff(fact_3370_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,
    ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList),Summary2)),Uu) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
tff(fact_3371_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,
    ! [A2: $o,B2: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,vEBT_Leaf((A2),(B2))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
tff(fact_3372_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,Summary2: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,TreeList),Summary2)),Xb) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
tff(fact_3373_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,Summary2: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList),Summary2)),Xb) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
tff(fact_3374_T__vebt__buildupi__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U)) ) ).

% T_vebt_buildupi_univ
tff(fact_3375_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,Xb),Xaa) = Y )
     => ( ( ? [A: $o,B: $o] : Xb = vEBT_Leaf((A),(B))
         => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)
             => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)
               => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $ite(
                            Xaa = Mi2,
                            one_one_nat,
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                Xaa = Ma2,
                                one_one_nat,
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                    one_one_nat,
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                        one_one_nat,
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),
                                          $let(
                                            h: nat,
                                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat) )) )) )) )) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_3376_vebt__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A),
                    $ite(Xaa = one_one_nat,(B),$false) ) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)),Xaa)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)),Xaa)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)),Xaa)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa))
                       => $ite(
                            Xaa = Mi2,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_3377_vebt__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A),
                      $ite(Xaa = one_one_nat,(B),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2) )
                 => ( ~ (Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ( ~ (Y)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi2,
                              $true,
                              $ite(
                                Xaa = Ma2,
                                $true,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                  $false,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_3378_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A),
                    $ite(Xaa = one_one_nat,(B),$false) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv),Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv),Uw)),Xaa)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2)),Xaa))
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_3379_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A),
                      $ite(Xaa = one_one_nat,(B),$false) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2)),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_3380_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xb),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A),
                      $ite(Xaa = one_one_nat,(B),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),Xaa)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv),Uw) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv),Uw)),Xaa)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2),S2)),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_3381_vebt__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A),
                      $ite(Xaa = one_one_nat,(B),$false) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa))
                 => ~ $ite(
                        Xaa = Mi2,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_3382_T__vebt__buildupi__gq__0,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(vEBT_V441764108873111860ildupi,N)) ).

% T_vebt_buildupi_gq_0
tff(fact_3383_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,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv2),Uw2)),Xb) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
tff(fact_3384_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,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy),Uz)),Xb) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
tff(fact_3385_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,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)),Xb) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
tff(fact_3386_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,
    ! [A2: $o,B2: $o,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,vEBT_Leaf((A2),(B2))),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
        $ite(Xb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_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)
tff(fact_3387_member__bound__height,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,Tb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))) ) ).

% member_bound_height
tff(fact_3388_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,Summary2: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2)),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
        $ite(
          Xb = Mi,
          one_one_nat,
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
            $ite(
              Xb = Ma,
              one_one_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi),
                  one_one_nat,
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                    $ite(
                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xb),
                      one_one_nat,
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),
                        $let(
                          h: nat,
                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat) )) )) )) )) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_3389_Tb__T__vebt__buildupi_H_H,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,N)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(vEBT_VEBT_Tb2,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% Tb_T_vebt_buildupi''
tff(fact_3390_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,Xb),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw) )
               => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2) )
                 => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2),Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2) )
                   => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                              $ite(
                                Xaa = Mi2,
                                one_one_nat,
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    Xaa = Ma2,
                                    one_one_nat,
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                        one_one_nat,
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                            one_one_nat,
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),
                                              $let(
                                                h: nat,
                                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat) )) )) )) )) )) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_3391_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv)) )
             => ( ~ (Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv))),Xaa)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)),Xaa)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,Va3),Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,Va3),Vb2)),Xaa)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi2 )
                          | ( Xaa = Ma2 )
                          | $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)),Xaa)) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_3392_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv)) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv))),Xaa)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)),Xaa)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,Va3),Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,Va3),Vb2)),Xaa))
                   => ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)),Xaa))
                     => ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)),Xaa))
                       => $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_3393_space__2__pow__bound,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Tb))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N)),one_one_real))) ) ).

% space_2_pow_bound
tff(fact_3394_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,Va3),Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,Va3),Vb2)),Xaa))
               => ~ ( ( Xaa = Mi2 )
                    | ( Xaa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,V2),TreeList2),Vc2)),Xaa))
                 => ~ ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 )
                      | $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd2)),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_3395_t__buildup__cnt,axiom,
    ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8346862874174094_d_u_p,N))),aa_real_real(times_times_real(aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(N))),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))) ).

% t_buildup_cnt
tff(fact_3396_T__vebt__buildupi__cnt_H,axiom,
    ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V441764108873111860ildupi,N))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(N)))) ).

% T_vebt_buildupi_cnt'
tff(fact_3397_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_3398_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_3399_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_3400_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) )
    <=> ( M = N ) ) ).

% of_nat_eq_iff
tff(fact_3401_two__realpow__ge__two,axiom,
    ! [N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,N))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N)) ) ).

% two_realpow_ge_two
tff(fact_3402_count__buildup_H,axiom,
    ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(N))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))) ).

% count_buildup'
tff(fact_3403_space__cnt,axiom,
    ! [Tb: vEBT_VEBT] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Tb))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Tb))) ).

% space_cnt
tff(fact_3404_semiring__1__class_Oof__nat__0,axiom,
    aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,zero_zero_nat) = zero_z3563351764282998399l_num1 ).

% semiring_1_class.of_nat_0
tff(fact_3405_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% semiring_1_class.of_nat_0
tff(fact_3406_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% semiring_1_class.of_nat_0
tff(fact_3407_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% semiring_1_class.of_nat_0
tff(fact_3408_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% semiring_1_class.of_nat_0
tff(fact_3409_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).

% semiring_1_class.of_nat_0
tff(fact_3410_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_3411_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_3412_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_3413_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_3414_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_z3403309356797280102nteger = aa_nat_Code_integer(semiri4939895301339042750nteger,N) )
    <=> ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
tff(fact_3415_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M) = zero_zero_rat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3416_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = zero_zero_real )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3417_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = zero_zero_int )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3418_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = zero_zero_nat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3419_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = zero_z3403309356797280102nteger )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3420_of__nat__numeral,axiom,
    ! [N: num] : aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,aa_num_nat(numeral_numeral_nat,N)) = numera7442385471795722001l_num1(N) ).

% of_nat_numeral
tff(fact_3421_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,N)) = numeral_numeral_rat(N) ).

% of_nat_numeral
tff(fact_3422_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,N)) = numeral_numeral_real(N) ).

% of_nat_numeral
tff(fact_3423_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).

% of_nat_numeral
tff(fact_3424_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,N)) = aa_num_nat(numeral_numeral_nat,N) ).

% of_nat_numeral
tff(fact_3425_of__nat__numeral,axiom,
    ! [N: num] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_num_nat(numeral_numeral_nat,N)) = numera6620942414471956472nteger(N) ).

% of_nat_numeral
tff(fact_3426_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% of_nat_less_iff
tff(fact_3427_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% of_nat_less_iff
tff(fact_3428_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% of_nat_less_iff
tff(fact_3429_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% of_nat_less_iff
tff(fact_3430_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% of_nat_less_iff
tff(fact_3431_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% of_nat_le_iff
tff(fact_3432_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% of_nat_le_iff
tff(fact_3433_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% of_nat_le_iff
tff(fact_3434_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% of_nat_le_iff
tff(fact_3435_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% of_nat_le_iff
tff(fact_3436_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_add
tff(fact_3437_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_add
tff(fact_3438_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_add
tff(fact_3439_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_add
tff(fact_3440_of__nat__add,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% of_nat_add
tff(fact_3441_of__nat__1,axiom,
    aa_nat_uint32(semiri2565882477558803405uint32,one_one_nat) = one_one_uint32 ).

% of_nat_1
tff(fact_3442_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_3443_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_3444_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_3445_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_3446_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_3447_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3448_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3449_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3450_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3451_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,N) )
    <=> ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3452_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,N) = one_one_rat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3453_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,N) = one_one_real )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3454_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,N) = one_one_int )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3455_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,N) = one_one_nat )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3456_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,N) = one_one_Code_integer )
    <=> ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3457_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_mult
tff(fact_3458_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_mult
tff(fact_3459_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mult
tff(fact_3460_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_mult
tff(fact_3461_of__nat__mult,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% of_nat_mult
tff(fact_3462_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(semiri8010041392384452111omplex,M)),N) ).

% semiring_1_class.of_nat_power
tff(fact_3463_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,M)),N) ).

% semiring_1_class.of_nat_power
tff(fact_3464_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,M)),N) ).

% semiring_1_class.of_nat_power
tff(fact_3465_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),N) ).

% semiring_1_class.of_nat_power
tff(fact_3466_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),N) ).

% semiring_1_class.of_nat_power
tff(fact_3467_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(semiri8010041392384452111omplex,B2)),W) = aa_nat_complex(semiri8010041392384452111omplex,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3468_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W) = aa_nat_real(semiri5074537144036343181t_real,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3469_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( ( aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W) = aa_nat_int(semiri1314217659103216013at_int,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3470_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W) = aa_nat_nat(semiri1316708129612266289at_nat,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3471_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2)),W) = aa_nat_Code_integer(semiri4939895301339042750nteger,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3472_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Xb) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(semiri8010041392384452111omplex,B2)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3473_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Xb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3474_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Xb) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3475_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Xb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3476_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Xb) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3477_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3478_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3479_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3480_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3481_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3482_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_uint32(semiri2565882477558803405uint32,aa_nat_nat(suc,M)) = plus_plus_uint32(one_one_uint32,aa_nat_uint32(semiri2565882477558803405uint32,M)) ).

% of_nat_Suc
tff(fact_3483_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = plus_plus_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_3484_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M)) ).

% of_nat_Suc
tff(fact_3485_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% of_nat_Suc
tff(fact_3486_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M)) ).

% of_nat_Suc
tff(fact_3487_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,M)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),aa_nat_Code_integer(semiri4939895301339042750nteger,M)) ).

% of_nat_Suc
tff(fact_3488_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% of_nat_0_less_iff
tff(fact_3489_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% of_nat_0_less_iff
tff(fact_3490_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% of_nat_0_less_iff
tff(fact_3491_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% of_nat_0_less_iff
tff(fact_3492_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% of_nat_0_less_iff
tff(fact_3493_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(Xb)),N) = aa_nat_complex(semiri8010041392384452111omplex,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3494_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: nat] :
      ( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),N) = aa_nat_rat(semiri681578069525770553at_rat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3495_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),N) = aa_nat_real(semiri5074537144036343181t_real,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3496_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) = aa_nat_int(semiri1314217659103216013at_int,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3497_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N) = aa_nat_nat(semiri1316708129612266289at_nat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3498_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xb)),N) = aa_nat_Code_integer(semiri4939895301339042750nteger,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3499_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,N: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Y) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(Xb)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3500_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,N: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Y) = aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3501_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3502_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,N: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3503_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,N: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3504_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,N: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xb)),N) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3505_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3506_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3507_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3508_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3509_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3510_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3511_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3512_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3513_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3514_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3515_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3516_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2)),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3517_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3518_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3519_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3520_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3521_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3522_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3523_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3524_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3525_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(W)),aa_nat_real(semiri5074537144036343181t_real,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),N) ) ).

% numeral_less_real_of_nat_iff
tff(fact_3526_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),numeral_numeral_real(W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_3527_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(N)),aa_nat_real(semiri5074537144036343181t_real,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,N)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_3528_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3529_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3530_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Xb)),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3531_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3532_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3533_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),N)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3534_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),N)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3535_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3536_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3537_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3538_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3539_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3540_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3541_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3542_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3543_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),N)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3544_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3545_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),N)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3546_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3547_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3548_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(I)),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3549_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(I)),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3550_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(numeral_numeral_rat(I)),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3551_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3552_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),N)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3553_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(semiri681578069525770553at_rat,Xb)) ).

% mult_of_nat_commute
tff(fact_3554_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Xb)),Y) = aa_real_real(times_times_real(Y),aa_nat_real(semiri5074537144036343181t_real,Xb)) ).

% mult_of_nat_commute
tff(fact_3555_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,Xb)) ).

% mult_of_nat_commute
tff(fact_3556_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,Xb)) ).

% mult_of_nat_commute
tff(fact_3557_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)) ).

% mult_of_nat_commute
tff(fact_3558_of__nat__0__le__iff,axiom,
    ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% of_nat_0_le_iff
tff(fact_3559_of__nat__0__le__iff,axiom,
    ! [N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% of_nat_0_le_iff
tff(fact_3560_of__nat__0__le__iff,axiom,
    ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% of_nat_0_le_iff
tff(fact_3561_of__nat__0__le__iff,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_0_le_iff
tff(fact_3562_of__nat__0__le__iff,axiom,
    ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_0_le_iff
tff(fact_3563_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_3564_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_3565_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_3566_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_3567_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger) ).

% of_nat_less_0_iff
tff(fact_3568_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)) != zero_zero_rat ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3569_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)) != zero_zero_real ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3570_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) != zero_zero_int ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3571_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,N)) != zero_zero_nat ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3572_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,N)) != zero_z3403309356797280102nteger ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3573_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ).

% less_imp_of_nat_less
tff(fact_3574_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ).

% less_imp_of_nat_less
tff(fact_3575_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% less_imp_of_nat_less
tff(fact_3576_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ) ).

% less_imp_of_nat_less
tff(fact_3577_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ) ).

% less_imp_of_nat_less
tff(fact_3578_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,N))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% of_nat_less_imp_less
tff(fact_3579_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% of_nat_less_imp_less
tff(fact_3580_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% of_nat_less_imp_less
tff(fact_3581_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% of_nat_less_imp_less
tff(fact_3582_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ) ).

% of_nat_less_imp_less
tff(fact_3583_div__mult2__eq_H,axiom,
    ! [A2: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% div_mult2_eq'
tff(fact_3584_div__mult2__eq_H,axiom,
    ! [A2: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% div_mult2_eq'
tff(fact_3585_div__mult2__eq_H,axiom,
    ! [A2: code_integer,M: nat,N: nat] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% div_mult2_eq'
tff(fact_3586_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J)) ) ).

% of_nat_mono
tff(fact_3587_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,I)),aa_nat_Code_integer(semiri4939895301339042750nteger,J)) ) ).

% of_nat_mono
tff(fact_3588_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J)) ) ).

% of_nat_mono
tff(fact_3589_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J)) ) ).

% of_nat_mono
tff(fact_3590_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J)) ) ).

% of_nat_mono
tff(fact_3591_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_3592_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_3593_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_3594_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,K) != zero_z3563351764282998399l_num1 )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_3595_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,K) != zero_zero_rat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_3596_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,K) != zero_zero_real )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_3597_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,K) != zero_zero_int )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_3598_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,K) != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_3599_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,K) != zero_z3403309356797280102nteger )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_3600_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,N)) ) ) ).

% of_nat_diff
tff(fact_3601_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% of_nat_diff
tff(fact_3602_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ).

% of_nat_diff
tff(fact_3603_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ) ) ).

% of_nat_diff
tff(fact_3604_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ) ) ).

% of_nat_diff
tff(fact_3605_reals__Archimedean3,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ! [Y4: real] :
        ? [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N3)),Xb)) ) ).

% reals_Archimedean3
tff(fact_3606_real__of__nat__div4,axiom,
    ! [N: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),Xb))),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,Xb))) ).

% real_of_nat_div4
tff(fact_3607_nat__less__real__le,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,N)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M)) ) ).

% nat_less_real_le
tff(fact_3608_nat__le__real__less,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_3609_of__nat__less__two__power,axiom,
    ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_nat_rat(power_power_rat(numeral_numeral_rat(aa_num_num(bit0,one))),N)) ).

% of_nat_less_two_power
tff(fact_3610_of__nat__less__two__power,axiom,
    ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N)) ).

% of_nat_less_two_power
tff(fact_3611_of__nat__less__two__power,axiom,
    ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% of_nat_less_two_power
tff(fact_3612_of__nat__less__two__power,axiom,
    ! [N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ).

% of_nat_less_two_power
tff(fact_3613_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( ( N != zero_zero_nat )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,M))),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% inverse_of_nat_le
tff(fact_3614_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( ( N != zero_zero_nat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,M))),divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,N))) ) ) ).

% inverse_of_nat_le
tff(fact_3615_real__archimedian__rdiv__eq__0,axiom,
    ! [Xb: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M3)),Xb)),C) )
         => ( Xb = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_3616_real__of__nat__div2,axiom,
    ! [N: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,Xb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),Xb)))) ).

% real_of_nat_div2
tff(fact_3617_real__of__nat__div3,axiom,
    ! [N: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,Xb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),Xb)))),one_one_real) ).

% real_of_nat_div3
tff(fact_3618_Tb_H__cnt,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_VEBT_Tb2,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,vEBT_vebt_buildup(N)))) ).

% Tb'_cnt
tff(fact_3619_cnt__cnt__eq,axiom,
    ! [Tb: vEBT_VEBT] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Tb) = aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Tb)) ).

% cnt_cnt_eq
tff(fact_3620_t__build__cnt,axiom,
    ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8646137997579335489_i_l_d,N))),aa_real_real(times_times_real(aa_VEBT_VEBT_real(vEBT_VEBT_cnt,vEBT_vebt_buildup(N))),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))) ).

% t_build_cnt
tff(fact_3621_linear__plus__1__le__power,axiom,
    ! [Xb: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),Xb)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)),N)) ) ).

% linear_plus_1_le_power
tff(fact_3622_delete__bound__size__univ,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: real,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Tb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),U))))) ) ) ).

% delete_bound_size_univ
tff(fact_3623_TBOUND__vebt__predi,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat_o(time_T8353473612707095248on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_predi,Tb),Tib),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb)))) ).

% TBOUND_vebt_predi
tff(fact_3624_TBOUND__vebt__succi,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat_o(time_T8353473612707095248on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_succi,Tb),Tib),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb)))) ).

% TBOUND_vebt_succi
tff(fact_3625_buildup__build__time,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,N)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,N)) ).

% buildup_build_time
tff(fact_3626_TBOUND__vebt__minti,axiom,
    ! [Tb: vEBT_VEBTi] : aa_nat_o(time_T8353473612707095248on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Tb)),one_one_nat) ).

% TBOUND_vebt_minti
tff(fact_3627_TBOUND__vebt__maxti,axiom,
    ! [Tb: vEBT_VEBTi] : aa_nat_o(time_T8353473612707095248on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Tb)),one_one_nat) ).

% TBOUND_vebt_maxti
tff(fact_3628_TBOUND__replicate,axiom,
    ! [Xb: heap_T2636463487746394924on_nat,C: nat,N: nat] :
      ( aa_nat_o(time_T8353473612707095248on_nat(Xb),C)
     => aa_nat_o(time_T3808005469503390304on_nat(vEBT_V792416675989592002on_nat(N,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),N))) ) ).

% TBOUND_replicate
tff(fact_3629_TBOUND__replicate,axiom,
    ! [Xb: heap_Time_Heap_nat,C: nat,N: nat] :
      ( aa_nat_o(time_TBOUND_nat(Xb),C)
     => aa_nat_o(time_TBOUND_list_nat(vEBT_V7726092123322077554ei_nat(N,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),N))) ) ).

% TBOUND_replicate
tff(fact_3630_TBOUND__replicate,axiom,
    ! [Xb: heap_Time_Heap_o,C: nat,N: nat] :
      ( aa_nat_o(time_TBOUND_o(Xb),C)
     => aa_nat_o(time_TBOUND_list_o(vEBT_V2326993469660664182atei_o(N,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),N))) ) ).

% TBOUND_replicate
tff(fact_3631_TBOUND__replicate,axiom,
    ! [Xb: heap_T8145700208782473153_VEBTi,C: nat,N: nat] :
      ( aa_nat_o(time_T5737551269749752165_VEBTi(Xb),C)
     => aa_nat_o(time_T8149879359713347829_VEBTi(vEBT_V1859673955506687831_VEBTi(N,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),N))) ) ).

% TBOUND_replicate
tff(fact_3632_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_num_int(numeral_numeral_int,V) )
    <=> ( M = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_3633_delete__bound__size__univ_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: real,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Tb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),U)))) ) ) ).

% delete_bound_size_univ'
tff(fact_3634_height__double__log__univ__size,axiom,
    ! [U: real,Deg2: nat,Tb: vEBT_VEBT] :
      ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),Deg2) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),Deg2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),U)))) ) ) ).

% height_double_log_univ_size
tff(fact_3635_Abs__fnat__hom__0,axiom,
    zero_z3563351764282998399l_num1 = aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,zero_zero_nat) ).

% Abs_fnat_hom_0
tff(fact_3636_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% zle_int
tff(fact_3637_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ? [N3: nat] : K = aa_nat_int(semiri1314217659103216013at_int,N3) ) ).

% zero_le_imp_eq_int
tff(fact_3638_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ~ ! [N3: nat] : K != aa_nat_int(semiri1314217659103216013at_int,N3) ) ).

% nonneg_int_cases
tff(fact_3639_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z)
    <=> ? [N8: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,N8)) ) ).

% zle_iff_zadd
tff(fact_3640_zdiv__int,axiom,
    ! [A2: nat,B2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% zdiv_int
tff(fact_3641_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
    <=> ? [N8: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N8))) ) ).

% zless_iff_Suc_zadd
tff(fact_3642_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ? [N3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3)
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N3) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_3643_pos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ~ ! [N3: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3) ) ) ).

% pos_int_cases
tff(fact_3644_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_3645_zdiff__int__split,axiom,
    ! [P: fun_int_o,Xb: nat,Y: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Y)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Y))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_3646_word__unat__power,axiom,
    ! [N: nat] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N) = aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% word_unat_power
tff(fact_3647_pred__bound__size__univ_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: real,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d2,Tb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),U)))) ) ) ).

% pred_bound_size_univ'
tff(fact_3648_succ__bound__size__univ_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: real,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c2,Tb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),U)))) ) ) ).

% succ_bound_size_univ'
tff(fact_3649_member__bound__size__univ,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: real,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_m_e_m_b_e_r,Tb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),U))))) ) ) ).

% member_bound_size_univ
tff(fact_3650_Bolzano,axiom,
    ! [A2: real,B2: real,P: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( ! [A: real,B: real,C2: real] :
            ( aa_real_o(aa_real_fun_real_o(P,A),B)
           => ( aa_real_o(aa_real_fun_real_o(P,B),C2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C2)
                 => aa_real_o(aa_real_fun_real_o(P,A),C2) ) ) ) )
       => ( ! [X2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
               => ? [D5: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
                    & ! [A: real,B: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(B),A)),D5) )
                       => aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P,A2),B2) ) ) ) ).

% Bolzano
tff(fact_3651_log__pow__cancel,axiom,
    ! [A2: real,B2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_real(log(A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),B2)) = aa_nat_real(semiri5074537144036343181t_real,B2) ) ) ) ).

% log_pow_cancel
tff(fact_3652_Tb__T__vebt__buildupi,axiom,
    ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,N))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(vEBT_VEBT_Tb,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% Tb_T_vebt_buildupi
tff(fact_3653_zero__le__log__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(log(A2),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_3654_log__le__zero__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(A2),Xb)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_3655_one__le__log__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(log(A2),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),Xb) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_3656_log__le__one__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(A2),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),A2) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_3657_log__le__cancel__iff,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(A2),Xb)),aa_real_real(log(A2),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_3658_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% log2_of_power_le
tff(fact_3659_TBOUND__highi,axiom,
    ! [Xb: nat,N: nat] : aa_nat_o(time_TBOUND_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Xb),N)),one_one_nat) ).

% TBOUND_highi
tff(fact_3660_TBOUND__lowi,axiom,
    ! [Xb: nat,N: nat] : aa_nat_o(time_TBOUND_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Xb),N)),one_one_nat) ).

% TBOUND_lowi
tff(fact_3661_Tb__Tb_H,axiom,
    ! [Tb: nat] : aa_nat_int(vEBT_VEBT_Tb,Tb) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_VEBT_Tb2,Tb)) ).

% Tb_Tb'
tff(fact_3662_log__one,axiom,
    ! [A2: real] : aa_real_real(log(A2),one_one_real) = zero_zero_real ).

% log_one
tff(fact_3663_zero__less__log__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(log(A2),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_3664_log__less__zero__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(A2),Xb)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_3665_one__less__log__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(log(A2),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Xb) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_3666_log__less__one__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(A2),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),A2) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_3667_log__less__cancel__iff,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(A2),Xb)),aa_real_real(log(A2),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_3668_log__eq__one,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_real(log(A2),A2) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_3669_TBOUND__vebt__memberi,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat_o(time_TBOUND_o(aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(vEBT_V854960066525838166emberi,Tb),Tib),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb)))) ).

% TBOUND_vebt_memberi
tff(fact_3670_log__base__change,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_real(log(B2),Xb) = aa_real_real(divide_divide_real(aa_real_real(log(A2),Xb)),aa_real_real(log(A2),B2)) ) ) ) ).

% log_base_change
tff(fact_3671_log__of__power__eq,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(log(B2),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).

% log_of_power_eq
tff(fact_3672_less__log__of__power,axiom,
    ! [B2: real,N: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log(B2),M)) ) ) ).

% less_log_of_power
tff(fact_3673_log__mult,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(log(A2),aa_real_real(times_times_real(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(log(A2),Xb)),aa_real_real(log(A2),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_3674_le__log__of__power,axiom,
    ! [B2: real,N: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log(B2),M)) ) ) ).

% le_log_of_power
tff(fact_3675_log__divide,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(log(A2),aa_real_real(divide_divide_real(Xb),Y)) = aa_real_real(minus_minus_real(aa_real_real(log(A2),Xb)),aa_real_real(log(A2),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_3676_log__base__pow,axiom,
    ! [A2: real,N: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_real(log(aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),Xb) = aa_real_real(divide_divide_real(aa_real_real(log(A2),Xb)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% log_base_pow
tff(fact_3677_log__nat__power,axiom,
    ! [Xb: real,B2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(log(B2),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log(B2),Xb)) ) ) ).

% log_nat_power
tff(fact_3678_log2__of__power__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
     => ( aa_nat_real(semiri5074537144036343181t_real,N) = aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ).

% log2_of_power_eq
tff(fact_3679_log__of__power__less,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B2),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% log_of_power_less
tff(fact_3680_log__of__power__le,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B2),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% log_of_power_le
tff(fact_3681_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% less_log2_of_power
tff(fact_3682_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% le_log2_of_power
tff(fact_3683_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ).

% log2_of_power_less
tff(fact_3684_Tb__T__vebt__buildupi_H,axiom,
    ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(vEBT_V9176841429113362141ildupi,N)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(vEBT_VEBT_Tb,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% Tb_T_vebt_buildupi'
tff(fact_3685_TBOUND__buildupi,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(time_T5737551269749752165_VEBTi(aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ).

% TBOUND_buildupi
tff(fact_3686_Tbuildupi__buildupi_H,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,N)) = aa_nat_int(vEBT_V9176841429113362141ildupi,N) ).

% Tbuildupi_buildupi'
tff(fact_3687_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_3688_minNull__delete__time__bound,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(Tb,Xb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Tb),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ) ).

% minNull_delete_time_bound
tff(fact_3689_VEBT__internal_OTb_Osimps_I2_J,axiom,
    aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,zero_zero_nat)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb.simps(2)
tff(fact_3690_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb'.simps(2)
tff(fact_3691_not__min__Null__member,axiom,
    ! [Tb: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Tb)
     => ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Tb),X_1) ) ).

% not_min_Null_member
tff(fact_3692_min__Null__member,axiom,
    ! [Tb: vEBT_VEBT,Xb: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Tb)
     => ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tb),Xb) ) ).

% min_Null_member
tff(fact_3693_minminNull,axiom,
    ! [Tb: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(Tb) = none_nat )
     => aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Tb) ) ).

% minminNull
tff(fact_3694_minNullmin,axiom,
    ! [Tb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Tb)
     => ( vEBT_vebt_mint(Tb) = none_nat ) ) ).

% minNullmin
tff(fact_3695_minNull__delete__time__bound_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(Tb,Xb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Tb),Xb)),one_one_nat) ) ) ).

% minNull_delete_time_bound'
tff(fact_3696_vebt__memberi__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xb: nat,Tb: vEBT_VEBT] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(vEBT_vebt_memberi,Tib),Xb)),aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(vEBT_V854960066525838166emberi,Tb),Tib),Xb)) ).

% vebt_memberi_refines
tff(fact_3697_TBOUND__minNull,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Tb)
     => aa_nat_o(time_T5737551269749752165_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(vEBT_V3964819847710782039nserti,Tb),Tib),Xb)),one_one_nat) ) ).

% TBOUND_minNull
tff(fact_3698_TBOUND__vebt__inserti,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] :
      aa_nat_o(time_T5737551269749752165_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(vEBT_V3964819847710782039nserti,Tb),Tib),Xb)),
        $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Tb),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb))))) ).

% TBOUND_vebt_inserti
tff(fact_3699_VEBT__internal_OT__vebt__buildupi_H_Osimps_I1_J,axiom,
    aa_nat_int(vEBT_V9176841429113362141ildupi,zero_zero_nat) = one_one_int ).

% VEBT_internal.T_vebt_buildupi'.simps(1)
tff(fact_3700_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_3701_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux),Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_3702_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
    aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% VEBT_internal.T_vebt_buildupi'.simps(2)
tff(fact_3703_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( ! [Uv: $o] : Xb != vEBT_Leaf($true,(Uv))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_3704_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( ( Xb != vEBT_Leaf($false,$false) )
       => ~ ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2),Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_3705_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
      <=> (Y) )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv: $o] : Xb = vEBT_Leaf($true,(Uv))
           => (Y) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => (Y) )
           => ( ( ? [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2),Uy2)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_3706_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He8640631688564546883on_nat,F1: fun_op5783775701855621896on_nat,F2: fun_o_4049874799390722306on_nat,VEBTi: vEBT_VEBTi] : aa_Hea6597239020403563964on_nat(H,aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(F1,F2),VEBTi)) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_fun5534811901068199356on_nat(aTP_Lamm_ju(H),F1),aa_fun4373765338990682812on_nat(aTP_Lamm_jv(H),F2)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3707_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He5988046180898962017Heap_o,F1: fun_op5783775701855621896on_nat,F2: fun_o_4049874799390722306on_nat,VEBTi: vEBT_VEBTi] : aa_Hea662945835469240978Heap_o(H,aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(F1,F2),VEBTi)) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_fun836855539625404138Heap_o(aTP_Lamm_jw(H),F1),aa_fun4676301296891262398Heap_o(aTP_Lamm_jx(H),F2)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3708_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He2623223239018661208_VEBTi,F1: fun_op5783775701855621896on_nat,F2: fun_o_4049874799390722306on_nat,VEBTi: vEBT_VEBTi] : aa_Hea2832269215923769937_VEBTi(H,aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(F1,F2),VEBTi)) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_fun8651712798330174033_VEBTi(aTP_Lamm_jy(H),F1),aa_fun8772555301277947729_VEBTi(aTP_Lamm_jz(H),F2)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3709_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He6140748070426431971on_nat,F1: fun_op1763656958002597022Heap_o,F2: fun_o_1207901889202581752Heap_o,VEBTi: vEBT_VEBTi] : aa_Hea41712926108436044on_nat(H,aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(F1,F2),VEBTi)) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_fun878802566620225708on_nat(aTP_Lamm_ka(H),F1),aa_fun3614415378023021644on_nat(aTP_Lamm_kb(H),F2)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3710_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He6597989805181038017Heap_o,F1: fun_op1763656958002597022Heap_o,F2: fun_o_1207901889202581752Heap_o,VEBTi: vEBT_VEBTi] : aa_Hea4824394052061459970Heap_o(H,aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(F1,F2),VEBTi)) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_fun2115832704338266106Heap_o(aTP_Lamm_kc(H),F1),aa_fun3804294760359788590Heap_o(aTP_Lamm_kd(H),F2)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3711_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He6517945810935171576_VEBTi,F1: fun_op1763656958002597022Heap_o,F2: fun_o_1207901889202581752Heap_o,VEBTi: vEBT_VEBTi] : aa_Hea4465279032261640929_VEBTi(H,aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(F1,F2),VEBTi)) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_fun7082494938152799553_VEBTi(aTP_Lamm_ke(H),F1),aa_fun8320297987776248033_VEBTi(aTP_Lamm_kf(H),F2)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3712_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He1783186893022644846on_nat,F1: fun_op5127434927081926045_VEBTi,F2: fun_o_6972675622308511895_VEBTi,VEBTi: vEBT_VEBTi] : aa_Hea6259128614175166183on_nat(H,aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(F1,F2),VEBTi)) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_fun7829374822385380327on_nat(aTP_Lamm_kg(H),F1),aa_fun4045834922583578215on_nat(aTP_Lamm_kh(H),F2)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3713_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He4624006784692404214Heap_o,F1: fun_op5127434927081926045_VEBTi,F2: fun_o_6972675622308511895_VEBTi,VEBTi: vEBT_VEBTi] : aa_Hea7612170549154561063Heap_o(H,aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(F1,F2),VEBTi)) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_fun7400618842136392575Heap_o(aTP_Lamm_ki(H),F1),aa_fun3457972572965721299Heap_o(aTP_Lamm_kj(H),F2)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3714_VEBTi_Ocase__distrib,axiom,
    ! [H: fun_He8247729123535230979_VEBTi,F1: fun_op5127434927081926045_VEBTi,F2: fun_o_6972675622308511895_VEBTi,VEBTi: vEBT_VEBTi] : aa_Hea2545666323642607868_VEBTi(H,aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(F1,F2),VEBTi)) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_fun4118786628394255868_VEBTi(aTP_Lamm_kk(H),F1),aa_fun5654756598358634620_VEBTi(aTP_Lamm_kl(H),F2)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_3715_VEBT__internal_OT__vebt__buildupi_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V441764108873111860ildupi,zero_zero_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% VEBT_internal.T_vebt_buildupi.simps(1)
tff(fact_3716_VEBT__internal_OT__vebt__buildupi_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% VEBT_internal.T_vebt_buildupi.simps(2)
tff(fact_3717_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_VEBT_Tb2,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb'.simps(1)
tff(fact_3718_VEBT__internal_OTb_Osimps_I1_J,axiom,
    aa_nat_int(vEBT_VEBT_Tb,zero_zero_nat) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb.simps(1)
tff(fact_3719_del__x__mia,axiom,
    ! [Xb: nat,Mi: nat,Ma: nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( ( ( Xb = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
         => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2),Xb) = $let(
                xn: nat,
                xn:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2)))))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(
                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                    $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary2,h),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,xn),
                                    $ite(
                                      xn = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                      Ma ))),Deg2,newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,xn),
                                  $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),Deg2,newlist),Summary2) ) ) ),
                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2) ) ) ) ) ) ) ) ).

% del_x_mia
tff(fact_3720_del__x__mi__lets__in__minNull,axiom,
    ! [Xb: nat,Mi: nat,Ma: nat,Deg2: nat,Xn: nat,H: nat,Summary2: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( Xb = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                       => ( ( Sn = vEBT_vebt_delete(Summary2,H) )
                         => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xn),
                                      $ite(
                                        Xn = Ma,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(Sn),
                                          $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                        Ma ))),Deg2,Newlist),Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_3721_del__x__mi__lets__in,axiom,
    ! [Xb: nat,Mi: nat,Ma: nat,Deg2: nat,Xn: nat,H: nat,Summary2: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( Xb = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2),Xb) = $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary2,H),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xn),
                                      $ite(
                                        Xn = Ma,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(sn),
                                          $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                        Ma ))),Deg2,Newlist),sn) ),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xn),
                                    $ite(Xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Ma))),Deg2,Newlist),Summary2) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_3722_del__x__mi,axiom,
    ! [Xb: nat,Mi: nat,Ma: nat,Deg2: nat,Xn: nat,H: nat,Summary2: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
      ( ( ( Xb = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
                 => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2),Xb) = $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L),
                        $let(
                          newlist: list_VEBT_VEBT,
                          newlist:= list_u1324408373059187874T_VEBT(TreeList,H,newnode),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary2,H),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xn),
                                      $ite(
                                        Xn = Ma,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(sn),
                                          $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                        Ma ))),Deg2,newlist),sn) ),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xn),
                                    $ite(Xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,H)))),Ma))),Deg2,newlist),Summary2) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_3723_del__in__range,axiom,
    ! [Mi: nat,Xb: nat,Ma: nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Xb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
         => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2),Xb) = $let(
                xn: nat,
                xn:= 
                  $ite(Xb = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2)))))),Xb),
                $let(
                  minn: nat,
                  minn:= 
                    $ite(Xb = Mi,xn,Mi),
                  $let(
                    h: nat,
                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(
                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                      $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                        $let(
                          newlist: list_VEBT_VEBT,
                          newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary2,h),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,minn),
                                      $ite(
                                        xn = Ma,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(sn),
                                          $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                        Ma ))),Deg2,newlist),sn) ),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,minn),
                                    $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),Deg2,newlist),Summary2) ) ) ),
                      aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2) ) ) ) ) ) ) ) ) ).

% del_in_range
tff(fact_3724_del__x__not__mia,axiom,
    ! [Mi: nat,Xb: nat,Ma: nat,Deg2: nat,H: nat,L: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
               => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2),Xb) = $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeList,H,newnode),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary2,H),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),
                                    $ite(
                                      Xb = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                      Ma ))),Deg2,newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),
                                  $ite(Xb = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,H)))),Ma))),Deg2,newlist),Summary2) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_3725_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,Xb: nat,Ma: nat,Deg2: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary2: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
               => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                 => ( ( Sn = vEBT_vebt_delete(Summary2,H) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
                       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),
                                    $ite(
                                      Xb = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(Sn),
                                        $ite(maxs = none_nat,Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                      Ma ))),Deg2,Newlist),Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_3726_vebt__inserti__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xb: nat,Tb: vEBT_VEBT] : refine5565527176597971370_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(vEBT_vebt_inserti,Tib),Xb),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(vEBT_V3964819847710782039nserti,Tb),Tib),Xb)) ).

% vebt_inserti_refines
tff(fact_3727_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBT,J: nat,Xb: vEBT_VEBT] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,J,Xb),I) = nth_VEBT_VEBT(L,I) ) ) ).

% nth_update_invalid
tff(fact_3728_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,J: nat,Xb: vEBT_VEBTi] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,J,Xb),I) = nth_VEBT_VEBTi(L,I) ) ) ).

% nth_update_invalid
tff(fact_3729_nth__update__invalid,axiom,
    ! [I: nat,L: list_real,J: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( aa_nat_real(nth_real(list_update_real(L,J,Xb)),I) = aa_nat_real(nth_real(L),I) ) ) ).

% nth_update_invalid
tff(fact_3730_nth__update__invalid,axiom,
    ! [I: nat,L: list_o,J: nat,Xb: $o] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( aa_nat_o(nth_o(list_update_o(L,J,(Xb))),I)
      <=> aa_nat_o(nth_o(L),I) ) ) ).

% nth_update_invalid
tff(fact_3731_nth__update__invalid,axiom,
    ! [I: nat,L: list_nat,J: nat,Xb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
     => ( aa_nat_nat(nth_nat(list_update_nat(L,J,Xb)),I) = aa_nat_nat(nth_nat(L),I) ) ) ).

% nth_update_invalid
tff(fact_3732_nth__update__invalid,axiom,
    ! [I: nat,L: list_int,J: nat,Xb: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( aa_nat_int(nth_int(list_update_int(L,J,Xb)),I) = aa_nat_int(nth_int(L),I) ) ) ).

% nth_update_invalid
tff(fact_3733_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,Xb: nat,Ma: nat,Deg2: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
               => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                 => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
                     => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),
                                  $ite(Xb = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Ma))),Deg2,Newlist),Summary2) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_3734_del__x__mi__lets__in__not__minNull,axiom,
    ! [Xb: nat,Mi: nat,Ma: nat,Deg2: nat,Xn: nat,H: nat,Summary2: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( Xb = Mi )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                     => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode)
                       => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xn),
                                    $ite(Xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Ma))),Deg2,Newlist),Summary2) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_3735_del__x__not__mi,axiom,
    ! [Mi: nat,Xb: nat,Ma: nat,Deg2: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
               => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeList,H,Newnode) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
                   => ( vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2),Xb) = $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary2,H),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),
                                    $ite(
                                      Xb = Ma,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,the_nat(maxs)))))) ),
                                      Ma ))),Deg2,Newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),
                                  $ite(Xb = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(Newlist,H)))),Ma))),Deg2,Newlist),Summary2) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_3736_in__set__upd__eq,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,Xb: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6829681357464350627n_assn(L))
     => ( member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Y)))
      <=> ( ( Xb = Y )
          | ( member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(L))
            & ! [Y3: produc6575502325842934193n_assn] : member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Y3))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3737_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Xb: vEBT_VEBT,Y: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y)))
      <=> ( ( Xb = Y )
          | ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(L))
            & ! [Y3: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y3))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3738_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,Xb: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
      <=> ( ( Xb = Y )
          | ( member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(L))
            & ! [Y3: vEBT_VEBTi] : member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y3))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3739_in__set__upd__eq,axiom,
    ! [I: nat,L: list_real,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( aa_set_real_o(member_real(Xb),set_real2(list_update_real(L,I,Y)))
      <=> ( ( Xb = Y )
          | ( aa_set_real_o(member_real(Xb),set_real2(L))
            & ! [Y3: real] : aa_set_real_o(member_real(Xb),set_real2(list_update_real(L,I,Y3))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3740_in__set__upd__eq,axiom,
    ! [I: nat,L: list_o,Xb: $o,Y: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( member_o((Xb),set_o2(list_update_o(L,I,(Y))))
      <=> ( ( (Xb)
          <=> (Y) )
          | ( member_o((Xb),set_o2(L))
            & ! [Y3: $o] : member_o((Xb),set_o2(list_update_o(L,I,(Y3)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3741_in__set__upd__eq,axiom,
    ! [I: nat,L: list_nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
     => ( aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(L,I,Y)))
      <=> ( ( Xb = Y )
          | ( aa_set_nat_o(member_nat(Xb),set_nat2(L))
            & ! [Y3: nat] : aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(L,I,Y3))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3742_in__set__upd__eq,axiom,
    ! [I: nat,L: list_int,Xb: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( aa_set_int_o(member_int(Xb),set_int2(list_update_int(L,I,Y)))
      <=> ( ( Xb = Y )
          | ( aa_set_int_o(member_int(Xb),set_int2(L))
            & ! [Y3: int] : aa_set_int_o(member_int(Xb),set_int2(list_update_int(L,I,Y3))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3743_in__set__upd__cases,axiom,
    ! [Xb: produc6575502325842934193n_assn,L: list_P8527749157015355191n_assn,I: nat,Y: produc6575502325842934193n_assn] :
      ( member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6829681357464350627n_assn(L))
         => ( Xb != Y ) )
       => member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(L)) ) ) ).

% in_set_upd_cases
tff(fact_3744_in__set__upd__cases,axiom,
    ! [Xb: vEBT_VEBT,L: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
         => ( Xb != Y ) )
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3745_in__set__upd__cases,axiom,
    ! [Xb: vEBT_VEBTi,L: list_VEBT_VEBTi,I: nat,Y: vEBT_VEBTi] :
      ( member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
         => ( Xb != Y ) )
       => member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3746_in__set__upd__cases,axiom,
    ! [Xb: real,L: list_real,I: nat,Y: real] :
      ( aa_set_real_o(member_real(Xb),set_real2(list_update_real(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
         => ( Xb != Y ) )
       => aa_set_real_o(member_real(Xb),set_real2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3747_in__set__upd__cases,axiom,
    ! [Xb: $o,L: list_o,I: nat,Y: $o] :
      ( member_o((Xb),set_o2(list_update_o(L,I,(Y))))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
         => ( (Xb)
          <=> ~ (Y) ) )
       => member_o((Xb),set_o2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3748_in__set__upd__cases,axiom,
    ! [Xb: nat,L: list_nat,I: nat,Y: nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
         => ( Xb != Y ) )
       => aa_set_nat_o(member_nat(Xb),set_nat2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3749_in__set__upd__cases,axiom,
    ! [Xb: int,L: list_int,I: nat,Y: int] :
      ( aa_set_int_o(member_int(Xb),set_int2(list_update_int(L,I,Y)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
         => ( Xb != Y ) )
       => aa_set_int_o(member_int(Xb),set_int2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3750_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,Xb: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6829681357464350627n_assn(L))
     => ( member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Y)))
      <=> ( ( Xb = Y )
          | ! [Y3: produc6575502325842934193n_assn] : member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Y3))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3751_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Xb: vEBT_VEBT,Y: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y)))
      <=> ( ( Xb = Y )
          | ! [Y3: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y3))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3752_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,Xb: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y)))
      <=> ( ( Xb = Y )
          | ! [Y3: vEBT_VEBTi] : member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y3))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3753_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_real,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( aa_set_real_o(member_real(Xb),set_real2(list_update_real(L,I,Y)))
      <=> ( ( Xb = Y )
          | ! [Y3: real] : aa_set_real_o(member_real(Xb),set_real2(list_update_real(L,I,Y3))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3754_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_o,Xb: $o,Y: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( member_o((Xb),set_o2(list_update_o(L,I,(Y))))
      <=> ( ( (Xb)
          <=> (Y) )
          | ! [Y3: $o] : member_o((Xb),set_o2(list_update_o(L,I,(Y3)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3755_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
     => ( aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(L,I,Y)))
      <=> ( ( Xb = Y )
          | ! [Y3: nat] : aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(L,I,Y3))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3756_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_int,Xb: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( aa_set_int_o(member_int(Xb),set_int2(list_update_int(L,I,Y)))
      <=> ( ( Xb = Y )
          | ! [Y3: int] : aa_set_int_o(member_int(Xb),set_int2(list_update_int(L,I,Y3))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3757_nth__list__update_H,axiom,
    ! [L: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT,J: nat] :
      nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,I,Xb),J) = $ite(
        ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)) ),
        Xb,
        nth_VEBT_VEBT(L,J) ) ).

% nth_list_update'
tff(fact_3758_nth__list__update_H,axiom,
    ! [L: list_VEBT_VEBTi,I: nat,Xb: vEBT_VEBTi,J: nat] :
      nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,I,Xb),J) = $ite(
        ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)) ),
        Xb,
        nth_VEBT_VEBTi(L,J) ) ).

% nth_list_update'
tff(fact_3759_nth__list__update_H,axiom,
    ! [L: list_real,I: nat,Xb: real,J: nat] :
      aa_nat_real(nth_real(list_update_real(L,I,Xb)),J) = $ite(
        ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L)) ),
        Xb,
        aa_nat_real(nth_real(L),J) ) ).

% nth_list_update'
tff(fact_3760_nth__list__update_H,axiom,
    ! [L: list_o,I: nat,Xb: $o,J: nat] :
      ( aa_nat_o(nth_o(list_update_o(L,I,(Xb))),J)
    <=> $ite(
          ( ( I = J )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L)) ),
          (Xb),
          aa_nat_o(nth_o(L),J) ) ) ).

% nth_list_update'
tff(fact_3761_nth__list__update_H,axiom,
    ! [L: list_nat,I: nat,Xb: nat,J: nat] :
      aa_nat_nat(nth_nat(list_update_nat(L,I,Xb)),J) = $ite(
        ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L)) ),
        Xb,
        aa_nat_nat(nth_nat(L),J) ) ).

% nth_list_update'
tff(fact_3762_nth__list__update_H,axiom,
    ! [L: list_int,I: nat,Xb: int,J: nat] :
      aa_nat_int(nth_int(list_update_int(L,I,Xb)),J) = $ite(
        ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L)) ),
        Xb,
        aa_nat_int(nth_int(L),J) ) ).

% nth_list_update'
tff(fact_3763_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,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2)),Xb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xb) ),
        one_one_nat,
        $ite(
          ( ( Xb = Mi )
          & ( Xb = Ma ) ),
          one_one_nat,
          $let(
            xn: nat,
            xn:= 
              $ite(Xb = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2)))))),Xb),
            $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,nth_VEBT_VEBT(TreeList,h)),l)),
                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),l)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Summary2),h),one_one_nat)),
                  one_one_nat ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
tff(fact_3764_vebt__delete_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2),Xb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xb) ),
        aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2),
        $ite(
          ( ( Xb = Mi )
          & ( Xb = Ma ) ),
          aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2),
          $let(
            xn: nat,
            xn:= 
              $ite(Xb = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2)))))),Xb),
            $let(
              minn: nat,
              minn:= 
                $ite(Xb = Mi,xn,Mi),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                  $let(
                    newnode: vEBT_VEBT,
                    newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                    $let(
                      newlist: list_VEBT_VEBT,
                      newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                      $ite(
                        aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= vEBT_vebt_delete(Summary2,h),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,minn),
                                  $ite(
                                    xn = Ma,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                    Ma ))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist),sn) ),
                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,minn),
                                $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),newlist),Summary2) ) ) ),
                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_3765_vebt__delete_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xb,Xaa) = Y )
     => ( ! [A: $o,B: $o] :
            ( ( Xb = vEBT_Leaf((A),(B)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y != vEBT_Leaf($false,(B)) ) ) )
       => ( ! [A: $o] :
              ( ? [B: $o] : Xb = vEBT_Leaf((A),(B))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != vEBT_Leaf((A),$false) ) ) )
         => ( ! [A: $o,B: $o] :
                ( ( Xb = vEBT_Leaf((A),(B)) )
               => ( ? [N3: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N3))
                 => ( Y != vEBT_Leaf((A),(B)) ) ) )
           => ( ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2),Summary) )
                 => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2),Summary) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                   => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                     => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( Y != $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),
                              $ite(
                                ( ( Xaa = Mi2 )
                                & ( Xaa = Ma2 ) ),
                                aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                  $let(
                                    minn: nat,
                                    minn:= 
                                      $ite(Xaa = Mi2,xn,Mi2),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          newnode: vEBT_VEBT,
                                          newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                                          $let(
                                            newlist: list_VEBT_VEBT,
                                            newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode),
                                            $ite(
                                              aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                              $let(
                                                sn: vEBT_VEBT,
                                                sn:= vEBT_vebt_delete(Summary,h),
                                                aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,minn),
                                                        $ite(
                                                          xn = Ma2,
                                                          $let(
                                                            maxs: option_nat,
                                                            maxs:= vEBT_vebt_maxt(sn),
                                                            $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                                          Ma2 ))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist),sn) ),
                                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,minn),
                                                      $ite(xn = Ma2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist),Summary) ) ) ),
                                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_3766_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Xb),Xaa) = Y )
     => ( ( ? [A: $o,B: $o] : Xb = vEBT_Leaf((A),(B))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A: $o,B: $o] : Xb = vEBT_Leaf((A),(B))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A: $o,B: $o] : Xb = vEBT_Leaf((A),(B))
             => ( ? [N3: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N3))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2),Summary)
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,TreeList2),Summary)
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( Y != $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                              one_one_nat,
                              $ite(
                                ( ( Xaa = Mi2 )
                                & ( Xaa = Ma2 ) ),
                                one_one_nat,
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,nth_VEBT_VEBT(TreeList2,h)),l)),
                                          $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),l)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Summary),h),one_one_nat)),
                                        one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
tff(fact_3767_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Xb),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),zero_zero_nat)) ) ) )
         => ( ! [A: $o,B: $o] :
                ( ( Xb = vEBT_Leaf((A),(B)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A: $o,B: $o] :
                  ( ( Xb = vEBT_Leaf((A),(B)) )
                 => ! [N3: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,N3)))) ) ) )
             => ( ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2),Summary) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2),Summary)),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,TreeList2),Summary) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,TreeList2),Summary)),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                         => ( ( Y = $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                                  | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                                  one_one_nat,
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    one_one_nat,
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                      $let(
                                        l: nat,
                                        l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                        $let(
                                          h: nat,
                                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,nth_VEBT_VEBT(TreeList2,h)),l)),
                                              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),l)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_V1232361888498592333_e_t_e,Summary),h),one_one_nat)),
                                            one_one_nat ) ) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
tff(fact_3768_vebt__delete_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xb,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = vEBT_Leaf($false,(B)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),zero_zero_nat)) ) ) )
         => ( ! [A: $o,B: $o] :
                ( ( Xb = vEBT_Leaf((A),(B)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = vEBT_Leaf((A),$false) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A: $o,B: $o] :
                  ( ( Xb = vEBT_Leaf((A),(B)) )
                 => ! [N3: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
                     => ( ( Y = vEBT_Leaf((A),(B)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,N3)))) ) ) )
             => ( ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2),Summary) )
                   => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2),Summary) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2),Summary)),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                     => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,TrLst2),Smry2)),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                       => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                         => ( ( Y = $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                                  | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                      $let(
                                        minn: nat,
                                        minn:= 
                                          $ite(Xaa = Mi2,xn,Mi2),
                                        $let(
                                          h: nat,
                                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                            $let(
                                              newnode: vEBT_VEBT,
                                              newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                                              $let(
                                                newlist: list_VEBT_VEBT,
                                                newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode),
                                                $ite(
                                                  aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                                  $let(
                                                    sn: vEBT_VEBT,
                                                    sn:= vEBT_vebt_delete(Summary,h),
                                                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,minn),
                                                            $ite(
                                                              xn = Ma2,
                                                              $let(
                                                                maxs: option_nat,
                                                                maxs:= vEBT_vebt_maxt(sn),
                                                                $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,the_nat(maxs)))))) ),
                                                              Ma2 ))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist),sn) ),
                                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,minn),
                                                          $ite(xn = Ma2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(nth_VEBT_VEBT(newlist,h)))),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist),Summary) ) ) ),
                                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) ) ) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_3769_set__swap,axiom,
    ! [I: nat,Xs: list_P8527749157015355191n_assn,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6829681357464350627n_assn(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s6829681357464350627n_assn(Xs))
       => ( set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(list_u4534839942911652127n_assn(Xs,I,nth_Pr1769885009046257848n_assn(Xs,J)),J,nth_Pr1769885009046257848n_assn(Xs,I))) = set_Pr1139785259514867910n_assn(Xs) ) ) ) ).

% set_swap
tff(fact_3770_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,I))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_3771_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s7982070591426661849_VEBTi(Xs))
       => ( set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,nth_VEBT_VEBTi(Xs,J)),J,nth_VEBT_VEBTi(Xs,I))) = set_VEBT_VEBTi2(Xs) ) ) ) ).

% set_swap
tff(fact_3772_set__swap,axiom,
    ! [I: nat,Xs: list_real,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_real(Xs))
       => ( set_real2(list_update_real(list_update_real(Xs,I,aa_nat_real(nth_real(Xs),J)),J,aa_nat_real(nth_real(Xs),I))) = set_real2(Xs) ) ) ) ).

% set_swap
tff(fact_3773_set__swap,axiom,
    ! [I: nat,Xs: list_o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(Xs))
       => ( set_o2(list_update_o(list_update_o(Xs,I,aa_nat_o(nth_o(Xs),J)),J,aa_nat_o(nth_o(Xs),I))) = set_o2(Xs) ) ) ) ).

% set_swap
tff(fact_3774_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_nat(Xs))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_3775_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
       => ( set_int2(list_update_int(list_update_int(Xs,I,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_3776_insert__simp__excp,axiom,
    ! [Mi: nat,Deg2: nat,TreeList: list_VEBT_VEBT,Xb: nat,Ma: nat,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
         => ( ( Xb != Ma )
           => ( vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mi),Ma))),Deg2,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary2)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_3777_insert__simp__norm,axiom,
    ! [Xb: nat,Deg2: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary2: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2)
         => ( ( Xb != Ma )
           => ( vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Deg2,TreeList),Summary2),Xb) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Ma))),Deg2,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary2,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary2)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_3778_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_3779_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Xb: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xb),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_3780_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( aa_nat_real(nth_real(list_update_real(Xs,I,Xb)),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_3781_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I,(Xb))),I)
      <=> (Xb) ) ) ).

% nth_list_update_eq
tff(fact_3782_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xb)),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_3783_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xb)),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_3784_TBOUND__minNulli,axiom,
    ! [Tb: vEBT_VEBTi] : aa_nat_o(time_TBOUND_o(aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Tb)),one_one_nat) ).

% TBOUND_minNulli
tff(fact_3785_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_3786_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xb: vEBT_VEBTi] : size_s7982070591426661849_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xb)) = size_s7982070591426661849_VEBTi(Xs) ).

% length_list_update
tff(fact_3787_length__list__update,axiom,
    ! [Xs: list_real,I: nat,Xb: real] : size_size_list_real(list_update_real(Xs,I,Xb)) = size_size_list_real(Xs) ).

% length_list_update
tff(fact_3788_length__list__update,axiom,
    ! [Xs: list_o,I: nat,Xb: $o] : size_size_list_o(list_update_o(Xs,I,(Xb))) = size_size_list_o(Xs) ).

% length_list_update
tff(fact_3789_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,Xb: nat] : size_size_list_nat(list_update_nat(Xs,I,Xb)) = size_size_list_nat(Xs) ).

% length_list_update
tff(fact_3790_length__list__update,axiom,
    ! [Xs: list_int,I: nat,Xb: int] : size_size_list_int(list_update_int(Xs,I,Xb)) = size_size_list_int(Xs) ).

% length_list_update
tff(fact_3791_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] : list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),I)) = Xs ).

% list_update_id
tff(fact_3792_list__update__id,axiom,
    ! [Xs: list_int,I: nat] : list_update_int(Xs,I,aa_nat_int(nth_int(Xs),I)) = Xs ).

% list_update_id
tff(fact_3793_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat] : list_u1324408373059187874T_VEBT(Xs,I,nth_VEBT_VEBT(Xs,I)) = Xs ).

% list_update_id
tff(fact_3794_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat] : list_u6098035379799741383_VEBTi(Xs,I,nth_VEBT_VEBTi(Xs,I)) = Xs ).

% list_update_id
tff(fact_3795_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,Xb: nat] :
      ( ( I != J )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xb)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_3796_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_int,Xb: int] :
      ( ( I != J )
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xb)),J) = aa_nat_int(nth_int(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_3797_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ( I != J )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),J) = nth_VEBT_VEBT(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_3798_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBTi,Xb: vEBT_VEBTi] :
      ( ( I != J )
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xb),J) = nth_VEBT_VEBTi(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_3799_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)) ).

% max_Suc_Suc
tff(fact_3800_max__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),zero_zero_nat) = N ).

% max_0R
tff(fact_3801_max__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N) = N ).

% max_0L
tff(fact_3802_max__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),zero_zero_nat) = A2 ).

% max_nat.right_neutral
tff(fact_3803_max__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) )
    <=> ( ( A2 = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_3804_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),A2) = A2 ).

% max_nat.left_neutral
tff(fact_3805_max__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_3806_max__0__1_I3_J,axiom,
    ! [Xb: num] : ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,numera1916890842035813515d_enat(Xb)) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(3)
tff(fact_3807_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_Cod1783974652533396728nteger(ord_max_Code_integer(zero_z3403309356797280102nteger),numera6620942414471956472nteger(Xb)) = numera6620942414471956472nteger(Xb) ).

% max_0_1(3)
tff(fact_3808_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_real_real(ord_max_real(zero_zero_real),numeral_numeral_real(Xb)) = numeral_numeral_real(Xb) ).

% max_0_1(3)
tff(fact_3809_max__0__1_I3_J,axiom,
    ! [Xb: num] : ord_max_rat(zero_zero_rat,numeral_numeral_rat(Xb)) = numeral_numeral_rat(Xb) ).

% max_0_1(3)
tff(fact_3810_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(3)
tff(fact_3811_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),aa_num_int(numeral_numeral_int,Xb)) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(3)
tff(fact_3812_max__0__1_I4_J,axiom,
    ! [Xb: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(Xb),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(4)
tff(fact_3813_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_Cod1783974652533396728nteger(ord_max_Code_integer(numera6620942414471956472nteger(Xb)),zero_z3403309356797280102nteger) = numera6620942414471956472nteger(Xb) ).

% max_0_1(4)
tff(fact_3814_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_real_real(ord_max_real(numeral_numeral_real(Xb)),zero_zero_real) = numeral_numeral_real(Xb) ).

% max_0_1(4)
tff(fact_3815_max__0__1_I4_J,axiom,
    ! [Xb: num] : ord_max_rat(numeral_numeral_rat(Xb),zero_zero_rat) = numeral_numeral_rat(Xb) ).

% max_0_1(4)
tff(fact_3816_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xb)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(4)
tff(fact_3817_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,Xb)),zero_zero_int) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(4)
tff(fact_3818_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V)) = $ite(aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)),numera1916890842035813515d_enat(V),numera1916890842035813515d_enat(U)) ).

% max_number_of(1)
tff(fact_3819_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_Cod1783974652533396728nteger(ord_max_Code_integer(numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)),numera6620942414471956472nteger(V),numera6620942414471956472nteger(U)) ).

% max_number_of(1)
tff(fact_3820_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_ma8239519435860878689l_num1(numera7442385471795722001l_num1(U),numera7442385471795722001l_num1(V)) = $ite(aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(numera7442385471795722001l_num1(U)),numera7442385471795722001l_num1(V)),numera7442385471795722001l_num1(V),numera7442385471795722001l_num1(U)) ).

% max_number_of(1)
tff(fact_3821_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(ord_max_real(numeral_numeral_real(U)),numeral_numeral_real(V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(U)),numeral_numeral_real(V)),numeral_numeral_real(V),numeral_numeral_real(U)) ).

% max_number_of(1)
tff(fact_3822_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_max_rat(numeral_numeral_rat(U),numeral_numeral_rat(V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(U)),numeral_numeral_rat(V)),numeral_numeral_rat(V),numeral_numeral_rat(U)) ).

% max_number_of(1)
tff(fact_3823_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V),aa_num_nat(numeral_numeral_nat,U)) ).

% max_number_of(1)
tff(fact_3824_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(1)
tff(fact_3825_max__0__1_I1_J,axiom,
    aa_real_real(ord_max_real(zero_zero_real),one_one_real) = one_one_real ).

% max_0_1(1)
tff(fact_3826_max__0__1_I1_J,axiom,
    ord_max_rat(zero_zero_rat,one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_3827_max__0__1_I1_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),one_one_nat) = one_one_nat ).

% max_0_1(1)
tff(fact_3828_max__0__1_I1_J,axiom,
    ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,one_on7984719198319812577d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(1)
tff(fact_3829_max__0__1_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_3830_max__0__1_I1_J,axiom,
    aa_Cod1783974652533396728nteger(ord_max_Code_integer(zero_z3403309356797280102nteger),one_one_Code_integer) = one_one_Code_integer ).

% max_0_1(1)
tff(fact_3831_max__0__1_I2_J,axiom,
    aa_real_real(ord_max_real(one_one_real),zero_zero_real) = one_one_real ).

% max_0_1(2)
tff(fact_3832_max__0__1_I2_J,axiom,
    ord_max_rat(one_one_rat,zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_3833_max__0__1_I2_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),zero_zero_nat) = one_one_nat ).

% max_0_1(2)
tff(fact_3834_max__0__1_I2_J,axiom,
    ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat,zero_z5237406670263579293d_enat) = one_on7984719198319812577d_enat ).

% max_0_1(2)
tff(fact_3835_max__0__1_I2_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_3836_max__0__1_I2_J,axiom,
    aa_Cod1783974652533396728nteger(ord_max_Code_integer(one_one_Code_integer),zero_z3403309356797280102nteger) = one_one_Code_integer ).

% max_0_1(2)
tff(fact_3837_max__0__1_I5_J,axiom,
    ! [Xb: num] : ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(Xb)) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(5)
tff(fact_3838_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_Cod1783974652533396728nteger(ord_max_Code_integer(one_one_Code_integer),numera6620942414471956472nteger(Xb)) = numera6620942414471956472nteger(Xb) ).

% max_0_1(5)
tff(fact_3839_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_real_real(ord_max_real(one_one_real),numeral_numeral_real(Xb)) = numeral_numeral_real(Xb) ).

% max_0_1(5)
tff(fact_3840_max__0__1_I5_J,axiom,
    ! [Xb: num] : ord_max_rat(one_one_rat,numeral_numeral_rat(Xb)) = numeral_numeral_rat(Xb) ).

% max_0_1(5)
tff(fact_3841_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(5)
tff(fact_3842_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),aa_num_int(numeral_numeral_int,Xb)) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(5)
tff(fact_3843_max__0__1_I6_J,axiom,
    ! [Xb: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(Xb),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(6)
tff(fact_3844_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_Cod1783974652533396728nteger(ord_max_Code_integer(numera6620942414471956472nteger(Xb)),one_one_Code_integer) = numera6620942414471956472nteger(Xb) ).

% max_0_1(6)
tff(fact_3845_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_real_real(ord_max_real(numeral_numeral_real(Xb)),one_one_real) = numeral_numeral_real(Xb) ).

% max_0_1(6)
tff(fact_3846_max__0__1_I6_J,axiom,
    ! [Xb: num] : ord_max_rat(numeral_numeral_rat(Xb),one_one_rat) = numeral_numeral_rat(Xb) ).

% max_0_1(6)
tff(fact_3847_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(6)
tff(fact_3848_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,Xb)),one_one_int) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(6)
tff(fact_3849_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I)
     => ( list_u1324408373059187874T_VEBT(Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_3850_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xb: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s7982070591426661849_VEBTi(Xs)),I)
     => ( list_u6098035379799741383_VEBTi(Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_3851_list__update__beyond,axiom,
    ! [Xs: list_real,I: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Xs)),I)
     => ( list_update_real(Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_3852_list__update__beyond,axiom,
    ! [Xs: list_o,I: nat,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),I)
     => ( list_update_o(Xs,I,(Xb)) = Xs ) ) ).

% list_update_beyond
tff(fact_3853_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),I)
     => ( list_update_nat(Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_3854_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),I)
     => ( list_update_int(Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_3855_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : semiri4216267220026989637d_enat(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = ord_ma741700101516333627d_enat(semiri4216267220026989637d_enat(Xb),semiri4216267220026989637d_enat(Y)) ).

% of_nat_max
tff(fact_3856_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_real_real(ord_max_real(aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).

% of_nat_max
tff(fact_3857_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_3858_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).

% of_nat_max
tff(fact_3859_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Y)) ).

% of_nat_max
tff(fact_3860_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)) ).

% nat_add_max_right
tff(fact_3861_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Q2)) ).

% nat_add_max_left
tff(fact_3862_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).

% nat_mult_max_right
tff(fact_3863_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).

% nat_mult_max_left
tff(fact_3864_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),M) ).

% nat_minus_add_max
tff(fact_3865_neq__if__length__neq,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( size_size_list_real(Xs) != size_size_list_real(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_3866_neq__if__length__neq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) != size_size_list_o(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_3867_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) != size_size_list_nat(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_3868_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) != size_size_list_int(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_3869_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_real] : size_size_list_real(Xs2) = N ).

% Ex_list_of_length
tff(fact_3870_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_o] : size_size_list_o(Xs2) = N ).

% Ex_list_of_length
tff(fact_3871_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = N ).

% Ex_list_of_length
tff(fact_3872_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = N ).

% Ex_list_of_length
tff(fact_3873_length__induct,axiom,
    ! [P: fun_list_real_o,Xs: list_real] :
      ( ! [Xs2: list_real] :
          ( ! [Ys2: list_real] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_real(Ys2)),size_size_list_real(Xs2))
             => aa_list_real_o(P,Ys2) )
         => aa_list_real_o(P,Xs2) )
     => aa_list_real_o(P,Xs) ) ).

% length_induct
tff(fact_3874_length__induct,axiom,
    ! [P: fun_list_o_o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys2: list_o] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys2)),size_size_list_o(Xs2))
             => aa_list_o_o(P,Ys2) )
         => aa_list_o_o(P,Xs2) )
     => aa_list_o_o(P,Xs) ) ).

% length_induct
tff(fact_3875_length__induct,axiom,
    ! [P: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ys2)),size_size_list_nat(Xs2))
             => aa_list_nat_o(P,Ys2) )
         => aa_list_nat_o(P,Xs2) )
     => aa_list_nat_o(P,Xs) ) ).

% length_induct
tff(fact_3876_length__induct,axiom,
    ! [P: fun_list_int_o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs2))
             => aa_list_int_o(P,Ys2) )
         => aa_list_int_o(P,Xs2) )
     => aa_list_int_o(P,Xs) ) ).

% length_induct
tff(fact_3877_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I4) = nth_VEBT_VEBT(Ys,I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3878_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( Xs = Ys )
    <=> ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Ys) )
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s7982070591426661849_VEBTi(Xs))
           => ( nth_VEBT_VEBTi(Xs,I4) = nth_VEBT_VEBTi(Ys,I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3879_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_real(Xs) = size_size_list_real(Ys) )
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_real(Xs))
           => ( aa_nat_real(nth_real(Xs),I4) = aa_nat_real(nth_real(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3880_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I4)
            <=> aa_nat_o(nth_o(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3881_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I4) = aa_nat_nat(nth_nat(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3882_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I4) = aa_nat_int(nth_int(Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3883_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na6887940067898170868VEBT_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
         => ? [X_12: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I4),X_12) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I4),nth_VEBT_VEBT(Xs3,I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_3884_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na154763851189438853EBTi_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
         => ? [X_12: vEBT_VEBTi] : aa_VEBT_VEBTi_o(aa_nat4571447880046010350EBTi_o(P,I4),X_12) )
    <=> ? [Xs3: list_VEBT_VEBTi] :
          ( ( size_s7982070591426661849_VEBTi(Xs3) = K )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
             => aa_VEBT_VEBTi_o(aa_nat4571447880046010350EBTi_o(P,I4),nth_VEBT_VEBTi(Xs3,I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_3885_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_real_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
         => ? [X_12: real] : aa_real_o(aa_nat_fun_real_o(P,I4),X_12) )
    <=> ? [Xs3: list_real] :
          ( ( size_size_list_real(Xs3) = K )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
             => aa_real_o(aa_nat_fun_real_o(P,I4),aa_nat_real(nth_real(Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_3886_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_o_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
         => ? [X_12: $o] : aa_o_o(aa_nat_fun_o_o(P,I4),(X_12)) )
    <=> ? [Xs3: list_o] :
          ( ( size_size_list_o(Xs3) = K )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
             => aa_o_o(aa_nat_fun_o_o(P,I4),aa_nat_o(nth_o(Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_3887_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
         => ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I4),X_12) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
             => aa_nat_o(aa_nat_fun_nat_o(P,I4),aa_nat_nat(nth_nat(Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_3888_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
         => ? [X_12: int] : aa_int_o(aa_nat_fun_int_o(P,I4),X_12) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
             => aa_int_o(aa_nat_fun_int_o(P,I4),aa_nat_int(nth_int(Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_3889_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys,I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3890_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
           => ( nth_VEBT_VEBTi(Xs,I2) = nth_VEBT_VEBTi(Ys,I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3891_nth__equalityI,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( size_size_list_real(Xs) = size_size_list_real(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
           => ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3892_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I2)
            <=> aa_nat_o(nth_o(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3893_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3894_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_3895_all__nat__less,axiom,
    ! [N: nat,P: fun_nat_o] :
      ( ! [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M6),N)
         => aa_nat_o(P,M6) )
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))
         => aa_nat_o(P,X) ) ) ).

% all_nat_less
tff(fact_3896_ex__nat__less,axiom,
    ! [N: nat,P: fun_nat_o] :
      ( ? [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M6),N)
          & aa_nat_o(P,M6) )
    <=> ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))
          & aa_nat_o(P,X) ) ) ).

% ex_nat_less
tff(fact_3897_length__pos__if__in__set,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3898_length__pos__if__in__set,axiom,
    ! [Xb: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s6829681357464350627n_assn(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3899_length__pos__if__in__set,axiom,
    ! [Xb: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xb),set_real2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_real(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3900_length__pos__if__in__set,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( member_o((Xb),set_o2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_o(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3901_length__pos__if__in__set,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3902_length__pos__if__in__set,axiom,
    ! [Xb: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xb),set_int2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_int(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3903_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o] :
      ( ! [X: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X,set_VEBT_VEBTi2(Xs))
         => aa_VEBT_VEBTi_o(P,X) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s7982070591426661849_VEBTi(Xs))
         => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(Xs,I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_3904_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_3905_all__set__conv__all__nth,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P: fun_Pr3571187171380813162assn_o] :
      ( ! [X: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(Xs))
         => aa_Pro2491721552408928739assn_o(P,X) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6829681357464350627n_assn(Xs))
         => aa_Pro2491721552408928739assn_o(P,nth_Pr1769885009046257848n_assn(Xs,I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_3906_all__set__conv__all__nth,axiom,
    ! [Xs: list_real,P: fun_real_o] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(Xs))
         => aa_real_o(P,X) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_real(Xs))
         => aa_real_o(P,aa_nat_real(nth_real(Xs),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_3907_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: fun_o_o] :
      ( ! [X: $o] :
          ( member_o((X),set_o2(Xs))
         => aa_o_o(P,(X)) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_3908_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => aa_nat_o(P,X) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_3909_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_o] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(Xs))
         => aa_int_o(P,X) )
    <=> ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_3910_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o,Xb: vEBT_VEBTi] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
         => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(Xs,I2)) )
     => ( member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(Xs))
       => aa_VEBT_VEBTi_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_3911_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I2)) )
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(Xs))
       => aa_VEBT_VEBT_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_3912_all__nth__imp__all__set,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P: fun_Pr3571187171380813162assn_o,Xb: produc6575502325842934193n_assn] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6829681357464350627n_assn(Xs))
         => aa_Pro2491721552408928739assn_o(P,nth_Pr1769885009046257848n_assn(Xs,I2)) )
     => ( member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(Xs))
       => aa_Pro2491721552408928739assn_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_3913_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_o,Xb: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
         => aa_real_o(P,aa_nat_real(nth_real(Xs),I2)) )
     => ( aa_set_real_o(member_real(Xb),set_real2(Xs))
       => aa_real_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_3914_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: fun_o_o,Xb: $o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I2)) )
     => ( member_o((Xb),set_o2(Xs))
       => aa_o_o(P,(Xb)) ) ) ).

% all_nth_imp_all_set
tff(fact_3915_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Xb: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I2)) )
     => ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
       => aa_nat_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_3916_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_o,Xb: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I2)) )
     => ( aa_set_int_o(member_int(Xb),set_int2(Xs))
       => aa_int_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_3917_in__set__conv__nth,axiom,
    ! [Xb: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s7982070591426661849_VEBTi(Xs))
          & ( nth_VEBT_VEBTi(Xs,I4) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_3918_in__set__conv__nth,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6755466524823107622T_VEBT(Xs))
          & ( nth_VEBT_VEBT(Xs,I4) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_3919_in__set__conv__nth,axiom,
    ! [Xb: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_s6829681357464350627n_assn(Xs))
          & ( nth_Pr1769885009046257848n_assn(Xs,I4) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_3920_in__set__conv__nth,axiom,
    ! [Xb: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xb),set_real2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_real(Xs))
          & ( aa_nat_real(nth_real(Xs),I4) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_3921_in__set__conv__nth,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( member_o((Xb),set_o2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_o(Xs))
          & ( aa_nat_o(nth_o(Xs),I4)
          <=> (Xb) ) ) ) ).

% in_set_conv_nth
tff(fact_3922_in__set__conv__nth,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_nat(Xs))
          & ( aa_nat_nat(nth_nat(Xs),I4) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_3923_in__set__conv__nth,axiom,
    ! [Xb: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xb),set_int2(Xs))
    <=> ? [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),size_size_list_int(Xs))
          & ( aa_nat_int(nth_int(Xs),I4) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_3924_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs))
     => ( ! [X2: vEBT_VEBTi] :
            ( member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(Xs))
           => aa_VEBT_VEBTi_o(P,X2) )
       => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(Xs,N)) ) ) ).

% list_ball_nth
tff(fact_3925_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs))
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
           => aa_VEBT_VEBT_o(P,X2) )
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,N)) ) ) ).

% list_ball_nth
tff(fact_3926_list__ball__nth,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn,P: fun_Pr3571187171380813162assn_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s6829681357464350627n_assn(Xs))
     => ( ! [X2: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
           => aa_Pro2491721552408928739assn_o(P,X2) )
       => aa_Pro2491721552408928739assn_o(P,nth_Pr1769885009046257848n_assn(Xs,N)) ) ) ).

% list_ball_nth
tff(fact_3927_list__ball__nth,axiom,
    ! [N: nat,Xs: list_real,P: fun_real_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_real(Xs))
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),set_real2(Xs))
           => aa_real_o(P,X2) )
       => aa_real_o(P,aa_nat_real(nth_real(Xs),N)) ) ) ).

% list_ball_nth
tff(fact_3928_list__ball__nth,axiom,
    ! [N: nat,Xs: list_o,P: fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_o(Xs))
     => ( ! [X2: $o] :
            ( member_o((X2),set_o2(Xs))
           => aa_o_o(P,(X2)) )
       => aa_o_o(P,aa_nat_o(nth_o(Xs),N)) ) ) ).

% list_ball_nth
tff(fact_3929_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_nat(Xs))
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
           => aa_nat_o(P,X2) )
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),N)) ) ) ).

% list_ball_nth
tff(fact_3930_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P: fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_int(Xs))
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),set_int2(Xs))
           => aa_int_o(P,X2) )
       => aa_int_o(P,aa_nat_int(nth_int(Xs),N)) ) ) ).

% list_ball_nth
tff(fact_3931_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs))
     => member_VEBT_VEBTi(nth_VEBT_VEBTi(Xs,N),set_VEBT_VEBTi2(Xs)) ) ).

% nth_mem
tff(fact_3932_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(nth_VEBT_VEBT(Xs,N)),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_3933_nth__mem,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s6829681357464350627n_assn(Xs))
     => member7957490590177025114n_assn(nth_Pr1769885009046257848n_assn(Xs,N),set_Pr1139785259514867910n_assn(Xs)) ) ).

% nth_mem
tff(fact_3934_nth__mem,axiom,
    ! [N: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_real(Xs))
     => aa_set_real_o(member_real(aa_nat_real(nth_real(Xs),N)),set_real2(Xs)) ) ).

% nth_mem
tff(fact_3935_nth__mem,axiom,
    ! [N: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_o(Xs))
     => member_o(aa_nat_o(nth_o(Xs),N),set_o2(Xs)) ) ).

% nth_mem
tff(fact_3936_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),N)),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_3937_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_int(Xs))
     => aa_set_int_o(member_int(aa_nat_int(nth_int(Xs),N)),set_int2(Xs)) ) ).

% nth_mem
tff(fact_3938_set__update__memI,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn,Xb: produc6575502325842934193n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s6829681357464350627n_assn(Xs))
     => member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(Xs,N,Xb))) ) ).

% set_update_memI
tff(fact_3939_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,N,Xb))) ) ).

% set_update_memI
tff(fact_3940_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Xb: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs))
     => member_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,N,Xb))) ) ).

% set_update_memI
tff(fact_3941_set__update__memI,axiom,
    ! [N: nat,Xs: list_real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_real(Xs))
     => aa_set_real_o(member_real(Xb),set_real2(list_update_real(Xs,N,Xb))) ) ).

% set_update_memI
tff(fact_3942_set__update__memI,axiom,
    ! [N: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_o(Xs))
     => member_o((Xb),set_o2(list_update_o(Xs,N,(Xb)))) ) ).

% set_update_memI
tff(fact_3943_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(Xs,N,Xb))) ) ).

% set_update_memI
tff(fact_3944_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_int(Xs))
     => aa_set_int_o(member_int(Xb),set_int2(list_update_int(Xs,N,Xb))) ) ).

% set_update_memI
tff(fact_3945_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,Xb) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_3946_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Xb: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( ( list_u6098035379799741383_VEBTi(Xs,I,Xb) = Xs )
      <=> ( nth_VEBT_VEBTi(Xs,I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_3947_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( ( list_update_real(Xs,I,Xb) = Xs )
      <=> ( aa_nat_real(nth_real(Xs),I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_3948_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( ( list_update_o(Xs,I,(Xb)) = Xs )
      <=> ( aa_nat_o(nth_o(Xs),I)
        <=> (Xb) ) ) ) ).

% list_update_same_conv
tff(fact_3949_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( ( list_update_nat(Xs,I,Xb) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_3950_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( ( list_update_int(Xs,I,Xb) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_3951_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),J) = $ite(I = J,Xb,nth_VEBT_VEBT(Xs,J)) ) ) ).

% nth_list_update
tff(fact_3952_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Xb: vEBT_VEBTi,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xb),J) = $ite(I = J,Xb,nth_VEBT_VEBTi(Xs,J)) ) ) ).

% nth_list_update
tff(fact_3953_nth__list__update,axiom,
    ! [I: nat,Xs: list_real,Xb: real,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( aa_nat_real(nth_real(list_update_real(Xs,I,Xb)),J) = $ite(I = J,Xb,aa_nat_real(nth_real(Xs),J)) ) ) ).

% nth_list_update
tff(fact_3954_nth__list__update,axiom,
    ! [I: nat,Xs: list_o,Xb: $o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I,(Xb))),J)
      <=> $ite(I = J,(Xb),aa_nat_o(nth_o(Xs),J)) ) ) ).

% nth_list_update
tff(fact_3955_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,Xb: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xb)),J) = $ite(I = J,Xb,aa_nat_nat(nth_nat(Xs),J)) ) ) ).

% nth_list_update
tff(fact_3956_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,Xb: int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xb)),J) = $ite(I = J,Xb,aa_nat_int(nth_int(Xs),J)) ) ) ).

% nth_list_update
tff(fact_3957_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi),Mi,Xb),
        $let(
          h: nat,
          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
            & ~ ( ( Xb = Mi )
                | ( Xb = Ma ) ) ),
            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi),Xb,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary2,h),Summary2)),
            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_3958_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( ! [A: $o,B: $o] :
            ( ( Xb = vEBT_Leaf((A),(B)) )
           => ( Y != $ite(
                  Xaa = zero_zero_nat,
                  vEBT_Leaf($true,(B)),
                  $ite(Xaa = one_one_nat,vEBT_Leaf((A),$true),vEBT_Leaf((A),(B))) ) ) )
       => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2) )
             => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2) )
               => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary) )
                 => ( Y != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,Xaa),
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                              & ~ ( ( Xaa = Mi2 )
                                  | ( Xaa = Ma2 ) ) ),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,
                                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Xaa,Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList2,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),vEBT_vebt_insert(Summary,h),Summary)),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_3959_vebt__insert_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero_nat,
                      vEBT_Leaf($true,(B)),
                      $ite(Xaa = one_one_nat,vEBT_Leaf((A),$true),vEBT_Leaf((A),(B))) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),Xaa)) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2) )
               => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2)),Xaa)) ) )
           => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2) )
                 => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2)),Xaa)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary) )
                   => ( ( Y = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,Xaa),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                  & ~ ( ( Xaa = Mi2 )
                                      | ( Xaa = Ma2 ) ) ),
                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Xaa,Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma2))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList2,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),vEBT_vebt_insert(Summary,h),Summary)),
                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_3960_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Xb),Xaa) = Y )
     => ( ( ? [A: $o,B: $o] : Xb = vEBT_Leaf((A),(B))
         => ( Y != one_one_nat ) )
       => ( ( ? [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2)
           => ( Y != one_one_nat ) )
         => ( ( ? [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary)
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,Xaa),
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                              & ~ ( ( Xaa = Mi2 )
                                  | ( Xaa = Ma2 ) ) ),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Summary),h),one_one_nat)),
                              one_one_nat ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_3961_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,Summary2: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2)),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi),Mi,Xb),
        $let(
          h: nat,
          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
            & ~ ( ( Xb = Mi )
                | ( Xb = Ma ) ) ),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Summary2),h),one_one_nat)),
            one_one_nat ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_3962_vebt__buildupi__refines,axiom,
    ! [N: nat] : refine5565527176597971370_VEBTi(aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,N),aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,N)) ).

% vebt_buildupi_refines
tff(fact_3963_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,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
tff(fact_3964_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,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
tff(fact_3965_TBOUND__vebt__buildupi,axiom,
    ! [N: nat] : aa_nat_o(time_T5737551269749752165_VEBTi(aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,N)),aa_nat_nat(vEBT_V441764108873111860ildupi,N)) ).

% TBOUND_vebt_buildupi
tff(fact_3966_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,Q2) = Q2 ).

% max_enat_simps(3)
tff(fact_3967_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : ord_ma741700101516333627d_enat(Q2,zero_z5237406670263579293d_enat) = Q2 ).

% max_enat_simps(2)
tff(fact_3968_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,
    ! [A2: $o,B2: $o,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,vEBT_Leaf((A2),(B2))),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_3969_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,
    ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S)),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_3970_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,
    ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S)),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_3971_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,Summary2: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary2)),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_3972_insersimp_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Tb),X_1)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Tb),Y)),one_one_nat) ) ) ).

% insersimp'
tff(fact_3973_insertsimp_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,L: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Tb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Tb),L)),one_one_nat) ) ) ).

% insertsimp'
tff(fact_3974_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,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ( ( Xb != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va2: nat] : Xb != aa_nat_nat(suc,aa_nat_nat(suc,Va2)) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
tff(fact_3975_insert_H__bound__height,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Tb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb))) ) ).

% insert'_bound_height
tff(fact_3976_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,vEBT_Leaf((A2),(B2))) = one_one_nat ).

% VEBT_internal.cnt'.simps(1)
tff(fact_3977_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,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
tff(fact_3978_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,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
tff(fact_3979_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_space2,vEBT_Leaf((A2),(B2))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.space'.simps(1)
tff(fact_3980_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_space,vEBT_Leaf((A2),(B2))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.space.simps(1)
tff(fact_3981_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Xb),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Y = one_one_nat )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),Xaa)) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2)),Xaa)) ) )
           => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2)),Xaa)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,Xaa),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                  & ~ ( ( Xaa = Mi2 )
                                      | ( Xaa = Ma2 ) ) ),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t2,Summary),h),one_one_nat)),
                                  one_one_nat ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_3982_T__vebt__buildupi,axiom,
    ! [N: nat,H: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,N),H)),aa_nat_nat(vEBT_V441764108873111860ildupi,N)) ).

% T_vebt_buildupi
tff(fact_3983_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
     => ~ ! [N3: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N3)))),E2) ) ).

% nat_approx_posE
tff(fact_3984_nat__approx__posE,axiom,
    ! [E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
     => ~ ! [N3: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N3)))),E2) ) ).

% nat_approx_posE
tff(fact_3985_int__ops_I6_J,axiom,
    ! [A2: nat,B2: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2))) ).

% int_ops(6)
tff(fact_3986_max__less__iff__conj,axiom,
    ! [Xb: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(ord_ma741700101516333627d_enat(Xb,Y)),Z)
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Xb),Z)
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_3987_max__less__iff__conj,axiom,
    ! [Xb: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(ord_max_Code_integer(Xb),Y)),Z)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),Z)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_3988_max__less__iff__conj,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ord_max_real(Xb),Y)),Z)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_3989_max__less__iff__conj,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,ord_max_rat(Xb,Y)),Z)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_3990_max__less__iff__conj,axiom,
    ! [Xb: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y)),Z)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z)
        & aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_3991_max__less__iff__conj,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_3992_max__less__iff__conj,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_3993_max_Oabsorb4,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(A2),B2)
     => ( ord_ma741700101516333627d_enat(A2,B2) = B2 ) ) ).

% max.absorb4
tff(fact_3994_max_Oabsorb4,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B2)
     => ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2) = B2 ) ) ).

% max.absorb4
tff(fact_3995_max_Oabsorb4,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_real(ord_max_real(A2),B2) = B2 ) ) ).

% max.absorb4
tff(fact_3996_max_Oabsorb4,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( ord_max_rat(A2,B2) = B2 ) ) ).

% max.absorb4
tff(fact_3997_max_Oabsorb4,axiom,
    ! [A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2) = B2 ) ) ).

% max.absorb4
tff(fact_3998_max_Oabsorb4,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = B2 ) ) ).

% max.absorb4
tff(fact_3999_max_Oabsorb4,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) = B2 ) ) ).

% max.absorb4
tff(fact_4000_max_Oabsorb3,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B2),A2)
     => ( ord_ma741700101516333627d_enat(A2,B2) = A2 ) ) ).

% max.absorb3
tff(fact_4001_max_Oabsorb3,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B2),A2)
     => ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2) = A2 ) ) ).

% max.absorb3
tff(fact_4002_max_Oabsorb3,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => ( aa_real_real(ord_max_real(A2),B2) = A2 ) ) ).

% max.absorb3
tff(fact_4003_max_Oabsorb3,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => ( ord_max_rat(A2,B2) = A2 ) ) ).

% max.absorb3
tff(fact_4004_max_Oabsorb3,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2) = A2 ) ) ).

% max.absorb3
tff(fact_4005_max_Oabsorb3,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = A2 ) ) ).

% max.absorb3
tff(fact_4006_max_Oabsorb3,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) = A2 ) ) ).

% max.absorb3
tff(fact_4007_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( aa_num_num(bit0,X22) = aa_num_num(bit0,Y22) )
    <=> ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
tff(fact_4008_time__replicate,axiom,
    ! [Xb: heap_T5738788834812785303t_unit,C: nat,N: nat,H: heap_e7401611519738050253t_unit] :
      ( ! [H5: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_t4224138285095624986t_unit(Xb,H5)),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_t4781937132199089312t_unit(vEBT_V7483891112628345579t_unit(N,Xb),H)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),N))) ) ).

% time_replicate
tff(fact_4009_time__replicate,axiom,
    ! [Xb: heap_T8145700208782473153_VEBTi,C: nat,N: nat,H: heap_e7401611519738050253t_unit] :
      ( ! [H5: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(Xb,H5)),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_t3534373299052942712_VEBTi(vEBT_V1859673955506687831_VEBTi(N,Xb),H)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C)),N))) ) ).

% time_replicate
tff(fact_4010_max_Oabsorb1,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B2),A2)
     => ( ord_ma741700101516333627d_enat(A2,B2) = A2 ) ) ).

% max.absorb1
tff(fact_4011_max_Oabsorb1,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),A2)
     => ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2) = A2 ) ) ).

% max.absorb1
tff(fact_4012_max_Oabsorb1,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( ord_max_rat(A2,B2) = A2 ) ) ).

% max.absorb1
tff(fact_4013_max_Oabsorb1,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2) = A2 ) ) ).

% max.absorb1
tff(fact_4014_max_Oabsorb1,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = A2 ) ) ).

% max.absorb1
tff(fact_4015_max_Oabsorb1,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) = A2 ) ) ).

% max.absorb1
tff(fact_4016_max_Oabsorb2,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A2),B2)
     => ( ord_ma741700101516333627d_enat(A2,B2) = B2 ) ) ).

% max.absorb2
tff(fact_4017_max_Oabsorb2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2)
     => ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2) = B2 ) ) ).

% max.absorb2
tff(fact_4018_max_Oabsorb2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( ord_max_rat(A2,B2) = B2 ) ) ).

% max.absorb2
tff(fact_4019_max_Oabsorb2,axiom,
    ! [A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2) = B2 ) ) ).

% max.absorb2
tff(fact_4020_max_Oabsorb2,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = B2 ) ) ).

% max.absorb2
tff(fact_4021_max_Oabsorb2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) = B2 ) ) ).

% max.absorb2
tff(fact_4022_max_Obounded__iff,axiom,
    ! [B2: extended_enat,C: extended_enat,A2: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(ord_ma741700101516333627d_enat(B2,C)),A2)
    <=> ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B2),A2)
        & aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A2) ) ) ).

% max.bounded_iff
tff(fact_4023_max_Obounded__iff,axiom,
    ! [B2: code_integer,C: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(ord_max_Code_integer(B2),C)),A2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),A2)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),A2) ) ) ).

% max.bounded_iff
tff(fact_4024_max_Obounded__iff,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,ord_max_rat(B2,C)),A2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A2) ) ) ).

% max.bounded_iff
tff(fact_4025_max_Obounded__iff,axiom,
    ! [B2: num,C: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B2),C)),A2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A2) ) ) ).

% max.bounded_iff
tff(fact_4026_max_Obounded__iff,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C)),A2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A2) ) ) ).

% max.bounded_iff
tff(fact_4027_max_Obounded__iff,axiom,
    ! [B2: int,C: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C)),A2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A2) ) ) ).

% max.bounded_iff
tff(fact_4028_verit__la__disequality,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = B2 )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ).

% verit_la_disequality
tff(fact_4029_verit__la__disequality,axiom,
    ! [A2: num,B2: num] :
      ( ( A2 = B2 )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2) ) ).

% verit_la_disequality
tff(fact_4030_verit__la__disequality,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2) ) ).

% verit_la_disequality
tff(fact_4031_verit__la__disequality,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ).

% verit_la_disequality
tff(fact_4032_verit__comp__simplify1_I2_J,axiom,
    ! [A2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A2),A2) ).

% verit_comp_simplify1(2)
tff(fact_4033_verit__comp__simplify1_I2_J,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),A2) ).

% verit_comp_simplify1(2)
tff(fact_4034_verit__comp__simplify1_I2_J,axiom,
    ! [A2: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),A2) ).

% verit_comp_simplify1(2)
tff(fact_4035_verit__comp__simplify1_I2_J,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),A2) ).

% verit_comp_simplify1(2)
tff(fact_4036_verit__comp__simplify1_I2_J,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),A2) ).

% verit_comp_simplify1(2)
tff(fact_4037_verit__comp__simplify1_I1_J,axiom,
    ! [A2: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A2),A2) ).

% verit_comp_simplify1(1)
tff(fact_4038_verit__comp__simplify1_I1_J,axiom,
    ! [A2: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),A2) ).

% verit_comp_simplify1(1)
tff(fact_4039_verit__comp__simplify1_I1_J,axiom,
    ! [A2: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A2),A2) ).

% verit_comp_simplify1(1)
tff(fact_4040_verit__comp__simplify1_I1_J,axiom,
    ! [A2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),A2) ).

% verit_comp_simplify1(1)
tff(fact_4041_verit__comp__simplify1_I1_J,axiom,
    ! [A2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A2),A2) ).

% verit_comp_simplify1(1)
tff(fact_4042_verit__la__generic,axiom,
    ! [A2: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),Xb)
      | ( A2 = Xb )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),A2) ) ).

% verit_la_generic
tff(fact_4043_verit__comp__simplify1_I3_J,axiom,
    ! [B3: real,A4: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),A4)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A4),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_4044_verit__comp__simplify1_I3_J,axiom,
    ! [B3: rat,A4: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B3),A4)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A4),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_4045_verit__comp__simplify1_I3_J,axiom,
    ! [B3: num,A4: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B3),A4)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A4),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_4046_verit__comp__simplify1_I3_J,axiom,
    ! [B3: nat,A4: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B3),A4)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_4047_verit__comp__simplify1_I3_J,axiom,
    ! [B3: int,A4: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B3),A4)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B3) ) ).

% verit_comp_simplify1(3)
tff(fact_4048_verit__sum__simplify,axiom,
    ! [A2: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(A2,zero_z3563351764282998399l_num1) = A2 ).

% verit_sum_simplify
tff(fact_4049_verit__sum__simplify,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),zero_zero_real) = A2 ).

% verit_sum_simplify
tff(fact_4050_verit__sum__simplify,axiom,
    ! [A2: rat] : plus_plus_rat(A2,zero_zero_rat) = A2 ).

% verit_sum_simplify
tff(fact_4051_verit__sum__simplify,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),zero_zero_nat) = A2 ).

% verit_sum_simplify
tff(fact_4052_verit__sum__simplify,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),zero_zero_int) = A2 ).

% verit_sum_simplify
tff(fact_4053_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one != aa_num_num(bit0,X22) ).

% verit_eq_simplify(10)
tff(fact_4054_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X33: num] : aa_num_num(bit0,X22) != aa_num_num(bit1,X33) ).

% verit_eq_simplify(14)
tff(fact_4055_real__arch__simple,axiom,
    ! [Xb: real] :
    ? [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N3)) ).

% real_arch_simple
tff(fact_4056_real__arch__simple,axiom,
    ! [Xb: rat] :
    ? [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_nat_rat(semiri681578069525770553at_rat,N3)) ).

% real_arch_simple
tff(fact_4057_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_13: nat] : aa_nat_o(P,X_13)
       => ? [N3: nat] :
            ( ~ aa_nat_o(P,N3)
            & aa_nat_o(P,aa_nat_nat(suc,N3)) ) ) ) ).

% exists_least_lemma
tff(fact_4058_verit__eq__simplify_I12_J,axiom,
    ! [X33: num] : one != aa_num_num(bit1,X33) ).

% verit_eq_simplify(12)
tff(fact_4059_reals__Archimedean2,axiom,
    ! [Xb: rat] :
    ? [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_nat_rat(semiri681578069525770553at_rat,N3)) ).

% reals_Archimedean2
tff(fact_4060_reals__Archimedean2,axiom,
    ! [Xb: real] :
    ? [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N3)) ).

% reals_Archimedean2
tff(fact_4061_max_Omono,axiom,
    ! [C: extended_enat,A2: extended_enat,D: extended_enat,B2: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A2)
     => ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(D),B2)
       => aa_Extended_enat_o(ord_le2932123472753598470d_enat(ord_ma741700101516333627d_enat(C,D)),ord_ma741700101516333627d_enat(A2,B2)) ) ) ).

% max.mono
tff(fact_4062_max_Omono,axiom,
    ! [C: code_integer,A2: code_integer,D: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,D),B2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(ord_max_Code_integer(C),D)),aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2)) ) ) ).

% max.mono
tff(fact_4063_max_Omono,axiom,
    ! [C: rat,A2: rat,D: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,ord_max_rat(C,D)),ord_max_rat(A2,B2)) ) ) ).

% max.mono
tff(fact_4064_max_Omono,axiom,
    ! [C: num,A2: num,D: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,D),B2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,C),D)),aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2)) ) ) ).

% max.mono
tff(fact_4065_max_Omono,axiom,
    ! [C: nat,A2: nat,D: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)) ) ) ).

% max.mono
tff(fact_4066_max_Omono,axiom,
    ! [C: int,A2: int,D: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C),D)),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2)) ) ) ).

% max.mono
tff(fact_4067_max_OorderE,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B2),A2)
     => ( A2 = ord_ma741700101516333627d_enat(A2,B2) ) ) ).

% max.orderE
tff(fact_4068_max_OorderE,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),A2)
     => ( A2 = aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2) ) ) ).

% max.orderE
tff(fact_4069_max_OorderE,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( A2 = ord_max_rat(A2,B2) ) ) ).

% max.orderE
tff(fact_4070_max_OorderE,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
     => ( A2 = aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2) ) ) ).

% max.orderE
tff(fact_4071_max_OorderE,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
     => ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) ) ) ).

% max.orderE
tff(fact_4072_max_OorderE,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( A2 = aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) ) ) ).

% max.orderE
tff(fact_4073_max_OorderI,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( ( A2 = ord_ma741700101516333627d_enat(A2,B2) )
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(B2),A2) ) ).

% max.orderI
tff(fact_4074_max_OorderI,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),A2) ) ).

% max.orderI
tff(fact_4075_max_OorderI,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = ord_max_rat(A2,B2) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ).

% max.orderI
tff(fact_4076_max_OorderI,axiom,
    ! [A2: num,B2: num] :
      ( ( A2 = aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2) )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2) ) ).

% max.orderI
tff(fact_4077_max_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2) ) ).

% max.orderI
tff(fact_4078_max_OorderI,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ).

% max.orderI
tff(fact_4079_max_OboundedE,axiom,
    ! [B2: extended_enat,C: extended_enat,A2: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(ord_ma741700101516333627d_enat(B2,C)),A2)
     => ~ ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B2),A2)
         => ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A2) ) ) ).

% max.boundedE
tff(fact_4080_max_OboundedE,axiom,
    ! [B2: code_integer,C: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(ord_max_Code_integer(B2),C)),A2)
     => ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),A2)
         => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),A2) ) ) ).

% max.boundedE
tff(fact_4081_max_OboundedE,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,ord_max_rat(B2,C)),A2)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A2) ) ) ).

% max.boundedE
tff(fact_4082_max_OboundedE,axiom,
    ! [B2: num,C: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B2),C)),A2)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A2) ) ) ).

% max.boundedE
tff(fact_4083_max_OboundedE,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C)),A2)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A2) ) ) ).

% max.boundedE
tff(fact_4084_max_OboundedE,axiom,
    ! [B2: int,C: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C)),A2)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A2) ) ) ).

% max.boundedE
tff(fact_4085_max_OboundedI,axiom,
    ! [B2: extended_enat,A2: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B2),A2)
     => ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A2)
       => aa_Extended_enat_o(ord_le2932123472753598470d_enat(ord_ma741700101516333627d_enat(B2,C)),A2) ) ) ).

% max.boundedI
tff(fact_4086_max_OboundedI,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(ord_max_Code_integer(B2),C)),A2) ) ) ).

% max.boundedI
tff(fact_4087_max_OboundedI,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,ord_max_rat(B2,C)),A2) ) ) ).

% max.boundedI
tff(fact_4088_max_OboundedI,axiom,
    ! [B2: num,A2: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B2),C)),A2) ) ) ).

% max.boundedI
tff(fact_4089_max_OboundedI,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C)),A2) ) ) ).

% max.boundedI
tff(fact_4090_max_OboundedI,axiom,
    ! [B2: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C)),A2) ) ) ).

% max.boundedI
tff(fact_4091_max_Oorder__iff,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B2),A2)
    <=> ( A2 = ord_ma741700101516333627d_enat(A2,B2) ) ) ).

% max.order_iff
tff(fact_4092_max_Oorder__iff,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),A2)
    <=> ( A2 = aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2) ) ) ).

% max.order_iff
tff(fact_4093_max_Oorder__iff,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
    <=> ( A2 = ord_max_rat(A2,B2) ) ) ).

% max.order_iff
tff(fact_4094_max_Oorder__iff,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
    <=> ( A2 = aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2) ) ) ).

% max.order_iff
tff(fact_4095_max_Oorder__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
    <=> ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) ) ) ).

% max.order_iff
tff(fact_4096_max_Oorder__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
    <=> ( A2 = aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) ) ) ).

% max.order_iff
tff(fact_4097_max_Ocobounded1,axiom,
    ! [A2: extended_enat,B2: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(A2),ord_ma741700101516333627d_enat(A2,B2)) ).

% max.cobounded1
tff(fact_4098_max_Ocobounded1,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2)) ).

% max.cobounded1
tff(fact_4099_max_Ocobounded1,axiom,
    ! [A2: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),ord_max_rat(A2,B2)) ).

% max.cobounded1
tff(fact_4100_max_Ocobounded1,axiom,
    ! [A2: num,B2: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2)) ).

% max.cobounded1
tff(fact_4101_max_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)) ).

% max.cobounded1
tff(fact_4102_max_Ocobounded1,axiom,
    ! [A2: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2)) ).

% max.cobounded1
tff(fact_4103_max_Ocobounded2,axiom,
    ! [B2: extended_enat,A2: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(B2),ord_ma741700101516333627d_enat(A2,B2)) ).

% max.cobounded2
tff(fact_4104_max_Ocobounded2,axiom,
    ! [B2: code_integer,A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2)) ).

% max.cobounded2
tff(fact_4105_max_Ocobounded2,axiom,
    ! [B2: rat,A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),ord_max_rat(A2,B2)) ).

% max.cobounded2
tff(fact_4106_max_Ocobounded2,axiom,
    ! [B2: num,A2: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2)) ).

% max.cobounded2
tff(fact_4107_max_Ocobounded2,axiom,
    ! [B2: nat,A2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)) ).

% max.cobounded2
tff(fact_4108_max_Ocobounded2,axiom,
    ! [B2: int,A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2)) ).

% max.cobounded2
tff(fact_4109_le__max__iff__disj,axiom,
    ! [Z: extended_enat,Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),ord_ma741700101516333627d_enat(Xb,Y))
    <=> ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),Xb)
        | aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_4110_le__max__iff__disj,axiom,
    ! [Z: code_integer,Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),aa_Cod1783974652533396728nteger(ord_max_Code_integer(Xb),Y))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),Xb)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_4111_le__max__iff__disj,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),ord_max_rat(Xb,Y))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xb)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_4112_le__max__iff__disj,axiom,
    ! [Z: num,Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xb)
        | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_4113_le__max__iff__disj,axiom,
    ! [Z: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_4114_le__max__iff__disj,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_4115_max_Oabsorb__iff1,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(B2),A2)
    <=> ( ord_ma741700101516333627d_enat(A2,B2) = A2 ) ) ).

% max.absorb_iff1
tff(fact_4116_max_Oabsorb__iff1,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),A2)
    <=> ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2) = A2 ) ) ).

% max.absorb_iff1
tff(fact_4117_max_Oabsorb__iff1,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
    <=> ( ord_max_rat(A2,B2) = A2 ) ) ).

% max.absorb_iff1
tff(fact_4118_max_Oabsorb__iff1,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2) = A2 ) ) ).

% max.absorb_iff1
tff(fact_4119_max_Oabsorb__iff1,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = A2 ) ) ).

% max.absorb_iff1
tff(fact_4120_max_Oabsorb__iff1,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) = A2 ) ) ).

% max.absorb_iff1
tff(fact_4121_max_Oabsorb__iff2,axiom,
    ! [A2: extended_enat,B2: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(A2),B2)
    <=> ( ord_ma741700101516333627d_enat(A2,B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_4122_max_Oabsorb__iff2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2)
    <=> ( aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_4123_max_Oabsorb__iff2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
    <=> ( ord_max_rat(A2,B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_4124_max_Oabsorb__iff2,axiom,
    ! [A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_4125_max_Oabsorb__iff2,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_4126_max_Oabsorb__iff2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_4127_max_OcoboundedI1,axiom,
    ! [C: extended_enat,A2: extended_enat,B2: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),A2)
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),ord_ma741700101516333627d_enat(A2,B2)) ) ).

% max.coboundedI1
tff(fact_4128_max_OcoboundedI1,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2)) ) ).

% max.coboundedI1
tff(fact_4129_max_OcoboundedI1,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),ord_max_rat(A2,B2)) ) ).

% max.coboundedI1
tff(fact_4130_max_OcoboundedI1,axiom,
    ! [C: num,A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2)) ) ).

% max.coboundedI1
tff(fact_4131_max_OcoboundedI1,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)) ) ).

% max.coboundedI1
tff(fact_4132_max_OcoboundedI1,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2)) ) ).

% max.coboundedI1
tff(fact_4133_max_OcoboundedI2,axiom,
    ! [C: extended_enat,B2: extended_enat,A2: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),B2)
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(C),ord_ma741700101516333627d_enat(A2,B2)) ) ).

% max.coboundedI2
tff(fact_4134_max_OcoboundedI2,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2)) ) ).

% max.coboundedI2
tff(fact_4135_max_OcoboundedI2,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),ord_max_rat(A2,B2)) ) ).

% max.coboundedI2
tff(fact_4136_max_OcoboundedI2,axiom,
    ! [C: num,B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2)) ) ).

% max.coboundedI2
tff(fact_4137_max_OcoboundedI2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)) ) ).

% max.coboundedI2
tff(fact_4138_max_OcoboundedI2,axiom,
    ! [C: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2)) ) ).

% max.coboundedI2
tff(fact_4139_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B2: extended_enat,A2: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(C),B2)
     => aa_Extended_enat_o(ord_le72135733267957522d_enat(C),ord_ma741700101516333627d_enat(A2,B2)) ) ).

% max.strict_coboundedI2
tff(fact_4140_max_Ostrict__coboundedI2,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2)) ) ).

% max.strict_coboundedI2
tff(fact_4141_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(ord_max_real(A2),B2)) ) ).

% max.strict_coboundedI2
tff(fact_4142_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),ord_max_rat(A2,B2)) ) ).

% max.strict_coboundedI2
tff(fact_4143_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B2)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2)) ) ).

% max.strict_coboundedI2
tff(fact_4144_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)) ) ).

% max.strict_coboundedI2
tff(fact_4145_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2)) ) ).

% max.strict_coboundedI2
tff(fact_4146_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A2: extended_enat,B2: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(C),A2)
     => aa_Extended_enat_o(ord_le72135733267957522d_enat(C),ord_ma741700101516333627d_enat(A2,B2)) ) ).

% max.strict_coboundedI1
tff(fact_4147_max_Ostrict__coboundedI1,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2)) ) ).

% max.strict_coboundedI1
tff(fact_4148_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(ord_max_real(A2),B2)) ) ).

% max.strict_coboundedI1
tff(fact_4149_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),ord_max_rat(A2,B2)) ) ).

% max.strict_coboundedI1
tff(fact_4150_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),A2)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2)) ) ).

% max.strict_coboundedI1
tff(fact_4151_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)) ) ).

% max.strict_coboundedI1
tff(fact_4152_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2)) ) ).

% max.strict_coboundedI1
tff(fact_4153_max_Ostrict__order__iff,axiom,
    ! [B2: extended_enat,A2: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B2),A2)
    <=> ( ( A2 = ord_ma741700101516333627d_enat(A2,B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_4154_max_Ostrict__order__iff,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B2),A2)
    <=> ( ( A2 = aa_Cod1783974652533396728nteger(ord_max_Code_integer(A2),B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_4155_max_Ostrict__order__iff,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
    <=> ( ( A2 = aa_real_real(ord_max_real(A2),B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_4156_max_Ostrict__order__iff,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
    <=> ( ( A2 = ord_max_rat(A2,B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_4157_max_Ostrict__order__iff,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2)
    <=> ( ( A2 = aa_num_num(aa_num_fun_num_num(ord_max_num,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_4158_max_Ostrict__order__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
    <=> ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_4159_max_Ostrict__order__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
    <=> ( ( A2 = aa_int_int(aa_int_fun_int_int(ord_max_int,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_4160_max_Ostrict__boundedE,axiom,
    ! [B2: extended_enat,C: extended_enat,A2: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(ord_ma741700101516333627d_enat(B2,C)),A2)
     => ~ ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B2),A2)
         => ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(C),A2) ) ) ).

% max.strict_boundedE
tff(fact_4161_max_Ostrict__boundedE,axiom,
    ! [B2: code_integer,C: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(ord_max_Code_integer(B2),C)),A2)
     => ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B2),A2)
         => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),A2) ) ) ).

% max.strict_boundedE
tff(fact_4162_max_Ostrict__boundedE,axiom,
    ! [B2: real,C: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ord_max_real(B2),C)),A2)
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,C),A2) ) ) ).

% max.strict_boundedE
tff(fact_4163_max_Ostrict__boundedE,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,ord_max_rat(B2,C)),A2)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A2) ) ) ).

% max.strict_boundedE
tff(fact_4164_max_Ostrict__boundedE,axiom,
    ! [B2: num,C: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,B2),C)),A2)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,C),A2) ) ) ).

% max.strict_boundedE
tff(fact_4165_max_Ostrict__boundedE,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C)),A2)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A2) ) ) ).

% max.strict_boundedE
tff(fact_4166_max_Ostrict__boundedE,axiom,
    ! [B2: int,C: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B2),C)),A2)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C),A2) ) ) ).

% max.strict_boundedE
tff(fact_4167_less__max__iff__disj,axiom,
    ! [Z: extended_enat,Xb: extended_enat,Y: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Z),ord_ma741700101516333627d_enat(Xb,Y))
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Z),Xb)
        | aa_Extended_enat_o(ord_le72135733267957522d_enat(Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_4168_less__max__iff__disj,axiom,
    ! [Z: code_integer,Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),aa_Cod1783974652533396728nteger(ord_max_Code_integer(Xb),Y))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),Xb)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_4169_less__max__iff__disj,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(ord_max_real(Xb),Y))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xb)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_4170_less__max__iff__disj,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),ord_max_rat(Xb,Y))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xb)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_4171_less__max__iff__disj,axiom,
    ! [Z: num,Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Xb)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_4172_less__max__iff__disj,axiom,
    ! [Z: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_4173_less__max__iff__disj,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Xb)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_4174_max__def__raw,axiom,
    ! [X3: extended_enat,Xa: extended_enat] :
      ord_ma741700101516333627d_enat(X3,Xa) = $ite(aa_Extended_enat_o(ord_le2932123472753598470d_enat(X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_4175_max__def__raw,axiom,
    ! [X3: code_integer,Xa: code_integer] :
      aa_Cod1783974652533396728nteger(ord_max_Code_integer(X3),Xa) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_4176_max__def__raw,axiom,
    ! [X3: set_nat,Xa: set_nat] :
      ord_max_set_nat(X3,Xa) = $ite(aa_set_nat_o(ord_less_eq_set_nat(X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_4177_max__def__raw,axiom,
    ! [X3: rat,Xa: rat] :
      ord_max_rat(X3,Xa) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_4178_max__def__raw,axiom,
    ! [X3: num,Xa: num] :
      aa_num_num(aa_num_fun_num_num(ord_max_num,X3),Xa) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_4179_max__def__raw,axiom,
    ! [X3: nat,Xa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_4180_max__def__raw,axiom,
    ! [X3: int,Xa: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,X3),Xa) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_4181_int__ops_I3_J,axiom,
    ! [N: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,N)) = aa_num_int(numeral_numeral_int,N) ).

% int_ops(3)
tff(fact_4182_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_4183_nat__int__comparison_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)) ) ).

% nat_int_comparison(2)
tff(fact_4184_nat__int__comparison_I3_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)) ) ).

% nat_int_comparison(3)
tff(fact_4185_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_4186_nat__less__as__int,axiom,
    ! [X3: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_less_as_int
tff(fact_4187_nat__leq__as__int,axiom,
    ! [X3: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_leq_as_int
tff(fact_4188_ex__less__of__nat__mult,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ? [N3: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N3)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_4189_ex__less__of__nat__mult,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N3)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_4190_int__ops_I4_J,axiom,
    ! [A2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A2)),one_one_int) ).

% int_ops(4)
tff(fact_4191_int__Suc,axiom,
    ! [N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ).

% int_Suc
tff(fact_4192_heigt__uplog__rel,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ).

% heigt_uplog_rel
tff(fact_4193_succ__bound__size__univ,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: real,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Tb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),U))))) ) ) ).

% succ_bound_size_univ
tff(fact_4194_TBOUND__option__case,axiom,
    ! [Tb: option_nat,Fa: heap_Time_Heap_nat,Bnd: nat,F: fun_na3533371890617757541ap_nat,Bnd2: fun_nat_nat] :
      ( ( ( Tb = none_nat )
       => aa_nat_o(time_TBOUND_nat(Fa),Bnd) )
     => ( ! [X2: nat] :
            ( ( Tb = aa_nat_option_nat(some_nat,X2) )
           => aa_nat_o(time_TBOUND_nat(aa_nat5821310369566504398ap_nat(F,X2)),aa_nat_nat(Bnd2,X2)) )
       => aa_nat_o(time_TBOUND_nat(case_o6609685678014844897at_nat(Fa,F,Tb)),case_option_nat_nat(Bnd,Bnd2,Tb)) ) ) ).

% TBOUND_option_case
tff(fact_4195_TBOUND__option__case,axiom,
    ! [Tb: option_num,Fa: heap_Time_Heap_nat,Bnd: nat,F: fun_nu8176864466684091547ap_nat,Bnd2: fun_num_nat] :
      ( ( ( Tb = none_num )
       => aa_nat_o(time_TBOUND_nat(Fa),Bnd) )
     => ( ! [X2: num] :
            ( ( Tb = aa_num_option_num(some_num,X2) )
           => aa_nat_o(time_TBOUND_nat(aa_num1241430908778062596ap_nat(F,X2)),aa_num_nat(Bnd2,X2)) )
       => aa_nat_o(time_TBOUND_nat(case_o3167017464170623531at_num(Fa,F,Tb)),case_option_nat_num(Bnd,Bnd2,Tb)) ) ) ).

% TBOUND_option_case
tff(fact_4196_TBOUND__option__case,axiom,
    ! [Tb: option_nat,Fa: heap_Time_Heap_o,Bnd: nat,F: fun_na2343073368905879023Heap_o,Bnd2: fun_nat_nat] :
      ( ( ( Tb = none_nat )
       => aa_nat_o(time_TBOUND_o(Fa),Bnd) )
     => ( ! [X2: nat] :
            ( ( Tb = aa_nat_option_nat(some_nat,X2) )
           => aa_nat_o(time_TBOUND_o(aa_nat2910048432523430448Heap_o(F,X2)),aa_nat_nat(Bnd2,X2)) )
       => aa_nat_o(time_TBOUND_o(aa_opt6753654531493786624Heap_o(case_o6892868863119666303_o_nat(Fa,F),Tb)),case_option_nat_nat(Bnd,Bnd2,Tb)) ) ) ).

% TBOUND_option_case
tff(fact_4197_TBOUND__option__case,axiom,
    ! [Tb: option_num,Fa: heap_Time_Heap_o,Bnd: nat,F: fun_nu9127643932893349689Heap_o,Bnd2: fun_num_nat] :
      ( ( ( Tb = none_num )
       => aa_nat_o(time_TBOUND_o(Fa),Bnd) )
     => ( ! [X2: num] :
            ( ( Tb = aa_num_option_num(some_num,X2) )
           => aa_nat_o(time_TBOUND_o(aa_num471246959656125306Heap_o(F,X2)),aa_num_nat(Bnd2,X2)) )
       => aa_nat_o(time_TBOUND_o(case_o3450200649275444937_o_num(Fa,F,Tb)),case_option_nat_num(Bnd,Bnd2,Tb)) ) ) ).

% TBOUND_option_case
tff(fact_4198_TBOUND__option__case,axiom,
    ! [Tb: option_nat,Fa: heap_T8145700208782473153_VEBTi,Bnd: nat,F: fun_na7848093160509028170_VEBTi,Bnd2: fun_nat_nat] :
      ( ( ( Tb = none_nat )
       => aa_nat_o(time_T5737551269749752165_VEBTi(Fa),Bnd) )
     => ( ! [X2: nat] :
            ( ( Tb = aa_nat_option_nat(some_nat,X2) )
           => aa_nat_o(time_T5737551269749752165_VEBTi(aa_nat5408703056993926195_VEBTi(F,X2)),aa_nat_nat(Bnd2,X2)) )
       => aa_nat_o(time_T5737551269749752165_VEBTi(aa_opt6483592524530923107_VEBTi(case_o3780387683879180358Ti_nat(Fa,F),Tb)),case_option_nat_nat(Bnd,Bnd2,Tb)) ) ) ).

% TBOUND_option_case
tff(fact_4199_TBOUND__option__case,axiom,
    ! [Tb: option_num,Fa: heap_T8145700208782473153_VEBTi,Bnd: nat,F: fun_nu5987414290159654528_VEBTi,Bnd2: fun_num_nat] :
      ( ( ( Tb = none_num )
       => aa_nat_o(time_T5737551269749752165_VEBTi(Fa),Bnd) )
     => ( ! [X2: num] :
            ( ( Tb = aa_num_option_num(some_num,X2) )
           => aa_nat_o(time_T5737551269749752165_VEBTi(aa_num3548024186644552553_VEBTi(F,X2)),aa_num_nat(Bnd2,X2)) )
       => aa_nat_o(time_T5737551269749752165_VEBTi(case_o337719470034958992Ti_num(Fa,F,Tb)),case_option_nat_num(Bnd,Bnd2,Tb)) ) ) ).

% TBOUND_option_case
tff(fact_4200_TBOUND__option__case,axiom,
    ! [Tb: option_nat,Fa: heap_T2636463487746394924on_nat,Bnd: nat,F: fun_na5556145776640407093on_nat,Bnd2: fun_nat_nat] :
      ( ( ( Tb = none_nat )
       => aa_nat_o(time_T8353473612707095248on_nat(Fa),Bnd) )
     => ( ! [X2: nat] :
            ( ( Tb = aa_nat_option_nat(some_nat,X2) )
           => aa_nat_o(time_T8353473612707095248on_nat(aa_nat947733127081326238on_nat(F,X2)),aa_nat_nat(Bnd2,X2)) )
       => aa_nat_o(time_T8353473612707095248on_nat(case_o2256915875499652529at_nat(Fa,F,Tb)),case_option_nat_nat(Bnd,Bnd2,Tb)) ) ) ).

% TBOUND_option_case
tff(fact_4201_TBOUND__option__case,axiom,
    ! [Tb: option_num,Fa: heap_T2636463487746394924on_nat,Bnd: nat,F: fun_nu1337165112811199083on_nat,Bnd2: fun_num_nat] :
      ( ( ( Tb = none_num )
       => aa_nat_o(time_T8353473612707095248on_nat(Fa),Bnd) )
     => ( ! [X2: num] :
            ( ( Tb = aa_num_option_num(some_num,X2) )
           => aa_nat_o(time_T8353473612707095248on_nat(aa_num5952124500106894036on_nat(F,X2)),aa_num_nat(Bnd2,X2)) )
       => aa_nat_o(time_T8353473612707095248on_nat(case_o8037619698510206971at_num(Fa,F,Tb)),case_option_nat_num(Bnd,Bnd2,Tb)) ) ) ).

% TBOUND_option_case
tff(fact_4202_TBOUND__option__case,axiom,
    ! [Tb: option4927543243414619207at_nat,Fa: heap_Time_Heap_nat,Bnd: nat,F: fun_Pr7768898707182944990ap_nat,Bnd2: fun_Pr2602117396471556196at_nat] :
      ( ( ( Tb = none_P5556105721700978146at_nat )
       => aa_nat_o(time_TBOUND_nat(Fa),Bnd) )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( Tb = some_P7363390416028606310at_nat(X2) )
           => aa_nat_o(time_TBOUND_nat(aa_Pro1370750447752133975ap_nat(F,X2)),aa_Pro4243173770044397973at_nat(Bnd2,X2)) )
       => aa_nat_o(time_TBOUND_nat(case_o3959993630158478256at_nat(Fa,F,Tb)),case_o2098746482150326116at_nat(Bnd,Bnd2,Tb)) ) ) ).

% TBOUND_option_case
tff(fact_4203_TBOUND__option__case,axiom,
    ! [Tb: option4927543243414619207at_nat,Fa: heap_T8145700208782473153_VEBTi,Bnd: nat,F: fun_Pr45531988981583299_VEBTi,Bnd2: fun_Pr2602117396471556196at_nat] :
      ( ( ( Tb = none_P5556105721700978146at_nat )
       => aa_nat_o(time_T5737551269749752165_VEBTi(Fa),Bnd) )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( Tb = some_P7363390416028606310at_nat(X2) )
           => aa_nat_o(time_T5737551269749752165_VEBTi(aa_Pro1395359565518501564_VEBTi(F,X2)),aa_Pro4243173770044397973at_nat(Bnd2,X2)) )
       => aa_nat_o(time_T5737551269749752165_VEBTi(case_o1356590567247012107at_nat(Fa,F,Tb)),case_o2098746482150326116at_nat(Bnd,Bnd2,Tb)) ) ) ).

% TBOUND_option_case
tff(fact_4204_TBOUND__assert_H__bind__strong,axiom,
    ! [P: $o,M: heap_T2636463487746394924on_nat,Tb: nat] :
      ( ( (P)
       => aa_nat_o(time_T8353473612707095248on_nat(M),Tb) )
     => aa_nat_o(time_T8353473612707095248on_nat(heap_T5999496708990702694on_nat(refine_Imp_assert((P)),aTP_Lamm_km(M))),
          $ite((P),Tb,zero_zero_nat)) ) ).

% TBOUND_assert'_bind_strong
tff(fact_4205_TBOUND__assert_H__bind__strong,axiom,
    ! [P: $o,M: heap_Time_Heap_nat,Tb: nat] :
      ( ( (P)
       => aa_nat_o(time_TBOUND_nat(M),Tb) )
     => aa_nat_o(time_TBOUND_nat(heap_T3781436268274291734it_nat(refine_Imp_assert((P)),aTP_Lamm_kn(M))),
          $ite((P),Tb,zero_zero_nat)) ) ).

% TBOUND_assert'_bind_strong
tff(fact_4206_TBOUND__assert_H__bind__strong,axiom,
    ! [P: $o,M: heap_Time_Heap_o,Tb: nat] :
      ( ( (P)
       => aa_nat_o(time_TBOUND_o(M),Tb) )
     => aa_nat_o(time_TBOUND_o(heap_T7577940988442287570unit_o(refine_Imp_assert((P)),aTP_Lamm_ko(M))),
          $ite((P),Tb,zero_zero_nat)) ) ).

% TBOUND_assert'_bind_strong
tff(fact_4207_TBOUND__assert_H__bind__strong,axiom,
    ! [P: $o,M: heap_T8145700208782473153_VEBTi,Tb: nat] :
      ( ( (P)
       => aa_nat_o(time_T5737551269749752165_VEBTi(M),Tb) )
     => aa_nat_o(time_T5737551269749752165_VEBTi(heap_T7982501707604696571_VEBTi(refine_Imp_assert((P)),aTP_Lamm_kp(M))),
          $ite((P),Tb,zero_zero_nat)) ) ).

% TBOUND_assert'_bind_strong
tff(fact_4208_pred__bound__size__univ,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: real,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Tb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),U))))) ) ) ).

% pred_bound_size_univ
tff(fact_4209_insert__bound__size__univ,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: real,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Tb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),U))))) ) ) ).

% insert_bound_size_univ
tff(fact_4210_delete__correct,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(Tb,Xb)) = minus_minus_set_nat(vEBT_set_vebt(Tb),insert_nat(Xb,bot_bot_set_nat)) ) ) ).

% delete_correct
tff(fact_4211_delete__correct_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(Tb,Xb)) = minus_minus_set_nat(vEBT_VEBT_set_vebt(Tb),insert_nat(Xb,bot_bot_set_nat)) ) ) ).

% delete_correct'
tff(fact_4212_singleton__conv2,axiom,
    ! [A2: vEBT_VEBT] : collect_VEBT_VEBT(fequal_VEBT_VEBT(A2)) = insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT) ).

% singleton_conv2
tff(fact_4213_singleton__conv2,axiom,
    ! [A2: complex] : collect_complex(fequal_complex(A2)) = insert_complex(A2,bot_bot_set_complex) ).

% singleton_conv2
tff(fact_4214_singleton__conv2,axiom,
    ! [A2: product_prod_int_int] : collec213857154873943460nt_int(fequal6010348034681699726nt_int(A2)) = insert5033312907999012233nt_int(A2,bot_bo1796632182523588997nt_int) ).

% singleton_conv2
tff(fact_4215_singleton__conv2,axiom,
    ! [A2: real] : collect_real(aa_real_fun_real_o(fequal_real,A2)) = insert_real(A2,bot_bot_set_real) ).

% singleton_conv2
tff(fact_4216_singleton__conv2,axiom,
    ! [A2: nat] : collect_nat(aa_nat_fun_nat_o(fequal_nat,A2)) = insert_nat(A2,bot_bot_set_nat) ).

% singleton_conv2
tff(fact_4217_singleton__conv2,axiom,
    ! [A2: int] : collect_int(aa_int_fun_int_o(fequal_int,A2)) = insert_int(A2,bot_bot_set_int) ).

% singleton_conv2
tff(fact_4218_singleton__conv,axiom,
    ! [A2: vEBT_VEBT] : collect_VEBT_VEBT(aTP_Lamm_kq(A2)) = insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT) ).

% singleton_conv
tff(fact_4219_singleton__conv,axiom,
    ! [A2: complex] : collect_complex(aTP_Lamm_kr(A2)) = insert_complex(A2,bot_bot_set_complex) ).

% singleton_conv
tff(fact_4220_singleton__conv,axiom,
    ! [A2: product_prod_int_int] : collec213857154873943460nt_int(aTP_Lamm_ks(A2)) = insert5033312907999012233nt_int(A2,bot_bo1796632182523588997nt_int) ).

% singleton_conv
tff(fact_4221_singleton__conv,axiom,
    ! [A2: real] : collect_real(aa_real_fun_real_o(aTP_Lamm_kt,A2)) = insert_real(A2,bot_bot_set_real) ).

% singleton_conv
tff(fact_4222_singleton__conv,axiom,
    ! [A2: nat] : collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ku,A2)) = insert_nat(A2,bot_bot_set_nat) ).

% singleton_conv
tff(fact_4223_singleton__conv,axiom,
    ! [A2: int] : collect_int(aa_int_fun_int_o(aTP_Lamm_kv,A2)) = insert_int(A2,bot_bot_set_int) ).

% singleton_conv
tff(fact_4224_ceiling__zero,axiom,
    archim2889992004027027881ng_rat(zero_zero_rat) = zero_zero_int ).

% ceiling_zero
tff(fact_4225_ceiling__zero,axiom,
    aa_real_int(archim7802044766580827645g_real,zero_zero_real) = zero_zero_int ).

% ceiling_zero
tff(fact_4226_ceiling__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,numeral_numeral_real(V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_4227_ceiling__numeral,axiom,
    ! [V: num] : archim2889992004027027881ng_rat(numeral_numeral_rat(V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_4228_ceiling__one,axiom,
    archim2889992004027027881ng_rat(one_one_rat) = one_one_int ).

% ceiling_one
tff(fact_4229_ceiling__one,axiom,
    aa_real_int(archim7802044766580827645g_real,one_one_real) = one_one_int ).

% ceiling_one
tff(fact_4230_ceiling__le__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% ceiling_le_zero
tff(fact_4231_ceiling__le__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xb)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat) ) ).

% ceiling_le_zero
tff(fact_4232_ceiling__le__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),numeral_numeral_real(V)) ) ).

% ceiling_le_numeral
tff(fact_4233_ceiling__le__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),numeral_numeral_rat(V)) ) ).

% ceiling_le_numeral
tff(fact_4234_zero__less__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb) ) ).

% zero_less_ceiling
tff(fact_4235_zero__less__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% zero_less_ceiling
tff(fact_4236_numeral__less__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(V)),Xb) ) ).

% numeral_less_ceiling
tff(fact_4237_numeral__less__ceiling,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(V)),Xb) ) ).

% numeral_less_ceiling
tff(fact_4238_ceiling__less__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% ceiling_less_one
tff(fact_4239_ceiling__less__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xb)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat) ) ).

% ceiling_less_one
tff(fact_4240_one__le__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb) ) ).

% one_le_ceiling
tff(fact_4241_one__le__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% one_le_ceiling
tff(fact_4242_ceiling__add__numeral,axiom,
    ! [Xb: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),numeral_numeral_real(V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_4243_ceiling__add__numeral,axiom,
    ! [Xb: rat,V: num] : archim2889992004027027881ng_rat(plus_plus_rat(Xb,numeral_numeral_rat(V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_4244_ceiling__le__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ).

% ceiling_le_one
tff(fact_4245_ceiling__le__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xb)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),one_one_rat) ) ).

% ceiling_le_one
tff(fact_4246_one__less__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ).

% one_less_ceiling
tff(fact_4247_one__less__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ).

% one_less_ceiling
tff(fact_4248_ceiling__add__one,axiom,
    ! [Xb: rat] : archim2889992004027027881ng_rat(plus_plus_rat(Xb,one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(Xb)),one_one_int) ).

% ceiling_add_one
tff(fact_4249_ceiling__add__one,axiom,
    ! [Xb: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int) ).

% ceiling_add_one
tff(fact_4250_ceiling__diff__numeral,axiom,
    ! [Xb: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(minus_minus_real(Xb),numeral_numeral_real(V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_4251_ceiling__diff__numeral,axiom,
    ! [Xb: rat,V: num] : archim2889992004027027881ng_rat(minus_minus_rat(Xb,numeral_numeral_rat(V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_4252_ceiling__numeral__power,axiom,
    ! [Xb: num,N: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),N)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) ).

% ceiling_numeral_power
tff(fact_4253_ceiling__numeral__power,axiom,
    ! [Xb: num,N: nat] : archim2889992004027027881ng_rat(aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),N)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) ).

% ceiling_numeral_power
tff(fact_4254_ceiling__diff__one,axiom,
    ! [Xb: rat] : archim2889992004027027881ng_rat(minus_minus_rat(Xb,one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,archim2889992004027027881ng_rat(Xb)),one_one_int) ).

% ceiling_diff_one
tff(fact_4255_ceiling__diff__one,axiom,
    ! [Xb: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(minus_minus_real(Xb),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int) ).

% ceiling_diff_one
tff(fact_4256_ceiling__less__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(minus_minus_real(numeral_numeral_real(V)),one_one_real)) ) ).

% ceiling_less_numeral
tff(fact_4257_ceiling__less__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),minus_minus_rat(numeral_numeral_rat(V),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_4258_numeral__le__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(numeral_numeral_real(V)),one_one_real)),Xb) ) ).

% numeral_le_ceiling
tff(fact_4259_numeral__le__ceiling,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(numeral_numeral_rat(V),one_one_rat)),Xb) ) ).

% numeral_le_ceiling
tff(fact_4260_insert__compr,axiom,
    ! [A2: vEBT_VEBT,B5: set_VEBT_VEBT] : insert_VEBT_VEBT(A2,B5) = collect_VEBT_VEBT(aa_set2497645856053385599VEBT_o(aTP_Lamm_kw(A2),B5)) ).

% insert_compr
tff(fact_4261_insert__compr,axiom,
    ! [A2: real,B5: set_real] : insert_real(A2,B5) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_kx(A2),B5)) ).

% insert_compr
tff(fact_4262_insert__compr,axiom,
    ! [A2: nat,B5: set_nat] : insert_nat(A2,B5) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ky(A2),B5)) ).

% insert_compr
tff(fact_4263_insert__compr,axiom,
    ! [A2: int,B5: set_int] : insert_int(A2,B5) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_kz(A2),B5)) ).

% insert_compr
tff(fact_4264_insert__compr,axiom,
    ! [A2: complex,B5: set_complex] : insert_complex(A2,B5) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_la(A2),B5)) ).

% insert_compr
tff(fact_4265_insert__compr,axiom,
    ! [A2: product_prod_int_int,B5: set_Pr958786334691620121nt_int] : insert5033312907999012233nt_int(A2,B5) = collec213857154873943460nt_int(aa_set2224860389584585407_int_o(aTP_Lamm_lb(A2),B5)) ).

% insert_compr
tff(fact_4266_insert__Collect,axiom,
    ! [A2: vEBT_VEBT,P: fun_VEBT_VEBT_o] : insert_VEBT_VEBT(A2,collect_VEBT_VEBT(P)) = collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_lc(A2),P)) ).

% insert_Collect
tff(fact_4267_insert__Collect,axiom,
    ! [A2: real,P: fun_real_o] : insert_real(A2,collect_real(P)) = collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_ld(A2),P)) ).

% insert_Collect
tff(fact_4268_insert__Collect,axiom,
    ! [A2: nat,P: fun_nat_o] : insert_nat(A2,collect_nat(P)) = collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_le(A2),P)) ).

% insert_Collect
tff(fact_4269_insert__Collect,axiom,
    ! [A2: int,P: fun_int_o] : insert_int(A2,collect_int(P)) = collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_lf(A2),P)) ).

% insert_Collect
tff(fact_4270_insert__Collect,axiom,
    ! [A2: complex,P: fun_complex_o] : insert_complex(A2,collect_complex(P)) = collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_lg(A2),P)) ).

% insert_Collect
tff(fact_4271_insert__Collect,axiom,
    ! [A2: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] : insert5033312907999012233nt_int(A2,collec213857154873943460nt_int(P)) = collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_lh(A2),P)) ).

% insert_Collect
tff(fact_4272_Collect__conv__if2,axiom,
    ! [A2: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_li(A2),P)) = $ite(aa_VEBT_VEBT_o(P,A2),insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT),bot_bo8194388402131092736T_VEBT) ).

% Collect_conv_if2
tff(fact_4273_Collect__conv__if2,axiom,
    ! [A2: complex,P: fun_complex_o] :
      collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_lj(A2),P)) = $ite(aa_complex_o(P,A2),insert_complex(A2,bot_bot_set_complex),bot_bot_set_complex) ).

% Collect_conv_if2
tff(fact_4274_Collect__conv__if2,axiom,
    ! [A2: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_lk(A2),P)) = $ite(aa_Pro4592068669816069595_int_o(P,A2),insert5033312907999012233nt_int(A2,bot_bo1796632182523588997nt_int),bot_bo1796632182523588997nt_int) ).

% Collect_conv_if2
tff(fact_4275_Collect__conv__if2,axiom,
    ! [A2: real,P: fun_real_o] :
      collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_ll(A2),P)) = $ite(aa_real_o(P,A2),insert_real(A2,bot_bot_set_real),bot_bot_set_real) ).

% Collect_conv_if2
tff(fact_4276_Collect__conv__if2,axiom,
    ! [A2: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_lm(A2),P)) = $ite(aa_nat_o(P,A2),insert_nat(A2,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if2
tff(fact_4277_Collect__conv__if2,axiom,
    ! [A2: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ln(A2),P)) = $ite(aa_int_o(P,A2),insert_int(A2,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if2
tff(fact_4278_Collect__conv__if,axiom,
    ! [A2: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_lo(A2),P)) = $ite(aa_VEBT_VEBT_o(P,A2),insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT),bot_bo8194388402131092736T_VEBT) ).

% Collect_conv_if
tff(fact_4279_Collect__conv__if,axiom,
    ! [A2: complex,P: fun_complex_o] :
      collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_lp(A2),P)) = $ite(aa_complex_o(P,A2),insert_complex(A2,bot_bot_set_complex),bot_bot_set_complex) ).

% Collect_conv_if
tff(fact_4280_Collect__conv__if,axiom,
    ! [A2: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_lq(A2),P)) = $ite(aa_Pro4592068669816069595_int_o(P,A2),insert5033312907999012233nt_int(A2,bot_bo1796632182523588997nt_int),bot_bo1796632182523588997nt_int) ).

% Collect_conv_if
tff(fact_4281_Collect__conv__if,axiom,
    ! [A2: real,P: fun_real_o] :
      collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_lr(A2),P)) = $ite(aa_real_o(P,A2),insert_real(A2,bot_bot_set_real),bot_bot_set_real) ).

% Collect_conv_if
tff(fact_4282_Collect__conv__if,axiom,
    ! [A2: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ls(A2),P)) = $ite(aa_nat_o(P,A2),insert_nat(A2,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if
tff(fact_4283_Collect__conv__if,axiom,
    ! [A2: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_lt(A2),P)) = $ite(aa_int_o(P,A2),insert_int(A2,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if
tff(fact_4284_time__assert_H,axiom,
    ! [P: $o,H: heap_e7401611519738050253t_unit] : time_t4224138285095624986t_unit(refine_Imp_assert((P)),H) = zero_zero_nat ).

% time_assert'
tff(fact_4285_ceiling__mono,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,Xb)) ) ).

% ceiling_mono
tff(fact_4286_ceiling__mono,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Y)),archim2889992004027027881ng_rat(Xb)) ) ).

% ceiling_mono
tff(fact_4287_ceiling__less__cancel,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xb)),archim2889992004027027881ng_rat(Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).

% ceiling_less_cancel
tff(fact_4288_ceiling__less__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim7802044766580827645g_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% ceiling_less_cancel
tff(fact_4289_ceiling__add__le,axiom,
    ! [Xb: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(plus_plus_rat(Xb,Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,archim2889992004027027881ng_rat(Xb)),archim2889992004027027881ng_rat(Y))) ).

% ceiling_add_le
tff(fact_4290_ceiling__add__le,axiom,
    ! [Xb: real,Y: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim7802044766580827645g_real,Y))) ).

% ceiling_add_le
tff(fact_4291_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) ).

% atLeast0_atMost_Suc
tff(fact_4292_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( insert_nat(M,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N) ) ) ).

% atLeastAtMost_insertL
tff(fact_4293_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N)) = insert_nat(aa_nat_nat(suc,N),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_4294_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N) = insert_nat(M,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_4295_remove__subset,axiom,
    ! [Xb: vEBT_VEBT,S3: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),S3)
     => aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT))),S3) ) ).

% remove_subset
tff(fact_4296_remove__subset,axiom,
    ! [Xb: real,S3: set_real] :
      ( aa_set_real_o(member_real(Xb),S3)
     => aa_set_real_o(ord_less_set_real(minus_minus_set_real(S3,insert_real(Xb,bot_bot_set_real))),S3) ) ).

% remove_subset
tff(fact_4297_remove__subset,axiom,
    ! [Xb: int,S3: set_int] :
      ( aa_set_int_o(member_int(Xb),S3)
     => aa_set_int_o(ord_less_set_int(minus_minus_set_int(S3,insert_int(Xb,bot_bot_set_int))),S3) ) ).

% remove_subset
tff(fact_4298_remove__subset,axiom,
    ! [Xb: nat,S3: set_nat] :
      ( aa_set_nat_o(member_nat(Xb),S3)
     => aa_set_nat_o(ord_less_set_nat(minus_minus_set_nat(S3,insert_nat(Xb,bot_bot_set_nat))),S3) ) ).

% remove_subset
tff(fact_4299_psubset__insert__iff,axiom,
    ! [A3: set_VEBT_VEBT,Xb: vEBT_VEBT,B5: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A3),insert_VEBT_VEBT(Xb,B5))
    <=> $ite(
          aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),B5),
          aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A3),B5),
          $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),A3),aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT))),B5),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A3),B5)) ) ) ).

% psubset_insert_iff
tff(fact_4300_psubset__insert__iff,axiom,
    ! [A3: set_real,Xb: real,B5: set_real] :
      ( aa_set_real_o(ord_less_set_real(A3),insert_real(Xb,B5))
    <=> $ite(
          aa_set_real_o(member_real(Xb),B5),
          aa_set_real_o(ord_less_set_real(A3),B5),
          $ite(aa_set_real_o(member_real(Xb),A3),aa_set_real_o(ord_less_set_real(minus_minus_set_real(A3,insert_real(Xb,bot_bot_set_real))),B5),aa_set_real_o(ord_less_eq_set_real(A3),B5)) ) ) ).

% psubset_insert_iff
tff(fact_4301_psubset__insert__iff,axiom,
    ! [A3: set_int,Xb: int,B5: set_int] :
      ( aa_set_int_o(ord_less_set_int(A3),insert_int(Xb,B5))
    <=> $ite(
          aa_set_int_o(member_int(Xb),B5),
          aa_set_int_o(ord_less_set_int(A3),B5),
          $ite(aa_set_int_o(member_int(Xb),A3),aa_set_int_o(ord_less_set_int(minus_minus_set_int(A3,insert_int(Xb,bot_bot_set_int))),B5),aa_set_int_o(ord_less_eq_set_int(A3),B5)) ) ) ).

% psubset_insert_iff
tff(fact_4302_psubset__insert__iff,axiom,
    ! [A3: set_nat,Xb: nat,B5: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A3),insert_nat(Xb,B5))
    <=> $ite(
          aa_set_nat_o(member_nat(Xb),B5),
          aa_set_nat_o(ord_less_set_nat(A3),B5),
          $ite(aa_set_nat_o(member_nat(Xb),A3),aa_set_nat_o(ord_less_set_nat(minus_minus_set_nat(A3,insert_nat(Xb,bot_bot_set_nat))),B5),aa_set_nat_o(ord_less_eq_set_nat(A3),B5)) ) ) ).

% psubset_insert_iff
tff(fact_4303_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,
    ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S)),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_4304_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,vEBT_Leaf((Uu),(Uv2))),zero_zero_nat) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
tff(fact_4305_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,N: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,vEBT_Leaf((Uv2),(Uw2))),aa_nat_nat(suc,N)) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
tff(fact_4306_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] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va)),Vb) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
tff(fact_4307_mult__ceiling__le,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(times_times_real(A2),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A2)),aa_real_int(archim7802044766580827645g_real,B2))) ) ) ).

% mult_ceiling_le
tff(fact_4308_mult__ceiling__le,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(aa_rat_rat(times_times_rat(A2),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,archim2889992004027027881ng_rat(A2)),archim2889992004027027881ng_rat(B2))) ) ) ).

% mult_ceiling_le
tff(fact_4309_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] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)),Va) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
tff(fact_4310_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,Xb: produc6575502325842934193n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6829681357464350627n_assn(L))
     => ( insert5290817439147925377n_assn(nth_Pr1769885009046257848n_assn(L,I),set_Pr1139785259514867910n_assn(list_u4534839942911652127n_assn(L,I,Xb))) = insert5290817439147925377n_assn(Xb,set_Pr1139785259514867910n_assn(L)) ) ) ).

% insert_swap_set_eq
tff(fact_4311_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( insert_VEBT_VEBT(nth_VEBT_VEBT(L,I),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Xb))) = insert_VEBT_VEBT(Xb,set_VEBT_VEBT2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_4312_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,Xb: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( insert_VEBT_VEBTi(nth_VEBT_VEBTi(L,I),set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Xb))) = insert_VEBT_VEBTi(Xb,set_VEBT_VEBTi2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_4313_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( insert_real(aa_nat_real(nth_real(L),I),set_real2(list_update_real(L,I,Xb))) = insert_real(Xb,set_real2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_4314_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( insert_o(aa_nat_o(nth_o(L),I),set_o2(list_update_o(L,I,(Xb)))) = insert_o((Xb),set_o2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_4315_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
     => ( insert_nat(aa_nat_nat(nth_nat(L),I),set_nat2(list_update_nat(L,I,Xb))) = insert_nat(Xb,set_nat2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_4316_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( insert_int(aa_nat_int(nth_int(L),I),set_int2(list_update_int(L,I,Xb))) = insert_int(Xb,set_int2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_4317_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,
    ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S)),Xb) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_4318_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Va: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,vEBT_Leaf((A2),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite((B2),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
tff(fact_4319_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,
    ! [A2: $o,B2: $o,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,vEBT_Leaf((A2),(B2))),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(Xb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_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)
tff(fact_4320_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] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(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)
tff(fact_4321_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,vEBT_Leaf((Uu),(B2))),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
tff(fact_4322_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] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(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)
tff(fact_4323_TBOUND__mono,axiom,
    ! [C: heap_T2636463487746394924on_nat,Tb: nat,T2: nat] :
      ( aa_nat_o(time_T8353473612707095248on_nat(C),Tb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Tb),T2)
       => aa_nat_o(time_T8353473612707095248on_nat(C),T2) ) ) ).

% TBOUND_mono
tff(fact_4324_TBOUND__mono,axiom,
    ! [C: heap_Time_Heap_nat,Tb: nat,T2: nat] :
      ( aa_nat_o(time_TBOUND_nat(C),Tb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Tb),T2)
       => aa_nat_o(time_TBOUND_nat(C),T2) ) ) ).

% TBOUND_mono
tff(fact_4325_TBOUND__mono,axiom,
    ! [C: heap_Time_Heap_o,Tb: nat,T2: nat] :
      ( aa_nat_o(time_TBOUND_o(C),Tb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Tb),T2)
       => aa_nat_o(time_TBOUND_o(C),T2) ) ) ).

% TBOUND_mono
tff(fact_4326_TBOUND__mono,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,Tb: nat,T2: nat] :
      ( aa_nat_o(time_T5737551269749752165_VEBTi(C),Tb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Tb),T2)
       => aa_nat_o(time_T5737551269749752165_VEBTi(C),T2) ) ) ).

% TBOUND_mono
tff(fact_4327_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A2: $o,Uw2: $o] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,vEBT_Leaf((A2),(Uw2))),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
tff(fact_4328_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] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)),Vj) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
tff(fact_4329_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] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)),Vi) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_4330_TBOUND__refines,axiom,
    ! [C: heap_Time_Heap_nat,Tb: nat,D: heap_Time_Heap_nat] :
      ( aa_nat_o(time_TBOUND_nat(C),Tb)
     => ( refine1365783493865988805es_nat(C,D)
       => aa_nat_o(time_TBOUND_nat(D),Tb) ) ) ).

% TBOUND_refines
tff(fact_4331_TBOUND__refines,axiom,
    ! [C: heap_T2636463487746394924on_nat,Tb: nat,D: heap_T2636463487746394924on_nat] :
      ( aa_nat_o(time_T8353473612707095248on_nat(C),Tb)
     => ( refine7594492741263601813on_nat(C,D)
       => aa_nat_o(time_T8353473612707095248on_nat(D),Tb) ) ) ).

% TBOUND_refines
tff(fact_4332_TBOUND__refines,axiom,
    ! [C: heap_Time_Heap_o,Tb: nat,D: heap_Time_Heap_o] :
      ( aa_nat_o(time_TBOUND_o(C),Tb)
     => ( aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(C),D)
       => aa_nat_o(time_TBOUND_o(D),Tb) ) ) ).

% TBOUND_refines
tff(fact_4333_TBOUND__refines,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,Tb: nat,D: heap_T8145700208782473153_VEBTi] :
      ( aa_nat_o(time_T5737551269749752165_VEBTi(C),Tb)
     => ( refine5565527176597971370_VEBTi(C,D)
       => aa_nat_o(time_T5737551269749752165_VEBTi(D),Tb) ) ) ).

% TBOUND_refines
tff(fact_4334_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,Summary2: vEBT_VEBT,Xb: nat] : aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList),Summary2)),Xb) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
tff(fact_4335_insersimp,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Y: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Tb),X_1)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Tb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ) ) ).

% insersimp
tff(fact_4336_insertsimp,axiom,
    ! [Tb: vEBT_VEBT,N: nat,L: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Tb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Tb),L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ) ) ).

% insertsimp
tff(fact_4337_TBOUND__return,axiom,
    ! [Xb: option_nat] : aa_nat_o(time_T8353473612707095248on_nat(heap_T3487192422709364219on_nat(Xb)),one_one_nat) ).

% TBOUND_return
tff(fact_4338_TBOUND__return,axiom,
    ! [Xb: nat] : aa_nat_o(time_TBOUND_nat(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Xb)),one_one_nat) ).

% TBOUND_return
tff(fact_4339_TBOUND__return,axiom,
    ! [Xb: $o] : aa_nat_o(time_TBOUND_o(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,(Xb))),one_one_nat) ).

% TBOUND_return
tff(fact_4340_TBOUND__return,axiom,
    ! [Xb: vEBT_VEBTi] : aa_nat_o(time_T5737551269749752165_VEBTi(aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Xb)),one_one_nat) ).

% TBOUND_return
tff(fact_4341_time__return,axiom,
    ! [Xb: product_unit,H: heap_e7401611519738050253t_unit] : time_t4224138285095624986t_unit(heap_T7507251653302230130t_unit(Xb),H) = one_one_nat ).

% time_return
tff(fact_4342_time__return,axiom,
    ! [Xb: option_nat,H: heap_e7401611519738050253t_unit] : time_time_option_nat(heap_T3487192422709364219on_nat(Xb),H) = one_one_nat ).

% time_return
tff(fact_4343_time__return,axiom,
    ! [Xb: nat,H: heap_e7401611519738050253t_unit] : time_time_nat(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Xb),H) = one_one_nat ).

% time_return
tff(fact_4344_time__return,axiom,
    ! [Xb: $o,H: heap_e7401611519738050253t_unit] : time_time_o(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,(Xb)),H) = one_one_nat ).

% time_return
tff(fact_4345_time__return,axiom,
    ! [Xb: vEBT_VEBTi,H: heap_e7401611519738050253t_unit] : time_time_VEBT_VEBTi(aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Xb),H) = one_one_nat ).

% time_return
tff(fact_4346_TBOUND__nth,axiom,
    ! [Xs: array_option_nat,I: nat] : aa_nat_o(time_T8353473612707095248on_nat(aa_nat947733127081326238on_nat(array_nth_option_nat(Xs),I)),one_one_nat) ).

% TBOUND_nth
tff(fact_4347_TBOUND__nth,axiom,
    ! [Xs: array_nat,I: nat] : aa_nat_o(time_TBOUND_nat(aa_nat5821310369566504398ap_nat(array_nth_nat(Xs),I)),one_one_nat) ).

% TBOUND_nth
tff(fact_4348_TBOUND__nth,axiom,
    ! [Xs: array_o,I: nat] : aa_nat_o(time_TBOUND_o(aa_nat2910048432523430448Heap_o(array_nth_o(Xs),I)),one_one_nat) ).

% TBOUND_nth
tff(fact_4349_TBOUND__nth,axiom,
    ! [Xs: array_VEBT_VEBTi,I: nat] : aa_nat_o(time_T5737551269749752165_VEBTi(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Xs),I)),one_one_nat) ).

% TBOUND_nth
tff(fact_4350_TBOUND__def,axiom,
    ! [M: heap_T5738788834812785303t_unit,Tb: nat] :
      ( aa_nat_o(time_T7469515765551943773t_unit(M),Tb)
    <=> ! [H6: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_t4224138285095624986t_unit(M,H6)),Tb) ) ).

% TBOUND_def
tff(fact_4351_TBOUND__def,axiom,
    ! [M: heap_T2636463487746394924on_nat,Tb: nat] :
      ( aa_nat_o(time_T8353473612707095248on_nat(M),Tb)
    <=> ! [H6: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_option_nat(M,H6)),Tb) ) ).

% TBOUND_def
tff(fact_4352_TBOUND__def,axiom,
    ! [M: heap_Time_Heap_nat,Tb: nat] :
      ( aa_nat_o(time_TBOUND_nat(M),Tb)
    <=> ! [H6: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_nat(M,H6)),Tb) ) ).

% TBOUND_def
tff(fact_4353_TBOUND__def,axiom,
    ! [M: heap_Time_Heap_o,Tb: nat] :
      ( aa_nat_o(time_TBOUND_o(M),Tb)
    <=> ! [H6: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_o(M,H6)),Tb) ) ).

% TBOUND_def
tff(fact_4354_TBOUND__def,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,Tb: nat] :
      ( aa_nat_o(time_T5737551269749752165_VEBTi(M),Tb)
    <=> ! [H6: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(M,H6)),Tb) ) ).

% TBOUND_def
tff(fact_4355_TBOUNDI,axiom,
    ! [M: heap_T5738788834812785303t_unit,Tb: nat] :
      ( ! [H5: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_t4224138285095624986t_unit(M,H5)),Tb)
     => aa_nat_o(time_T7469515765551943773t_unit(M),Tb) ) ).

% TBOUNDI
tff(fact_4356_TBOUNDI,axiom,
    ! [M: heap_T2636463487746394924on_nat,Tb: nat] :
      ( ! [H5: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_option_nat(M,H5)),Tb)
     => aa_nat_o(time_T8353473612707095248on_nat(M),Tb) ) ).

% TBOUNDI
tff(fact_4357_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_nat,Tb: nat] :
      ( ! [H5: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_nat(M,H5)),Tb)
     => aa_nat_o(time_TBOUND_nat(M),Tb) ) ).

% TBOUNDI
tff(fact_4358_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_o,Tb: nat] :
      ( ! [H5: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_o(M,H5)),Tb)
     => aa_nat_o(time_TBOUND_o(M),Tb) ) ).

% TBOUNDI
tff(fact_4359_TBOUNDI,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,Tb: nat] :
      ( ! [H5: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(M,H5)),Tb)
     => aa_nat_o(time_T5737551269749752165_VEBTi(M),Tb) ) ).

% TBOUNDI
tff(fact_4360_TBOUNDD,axiom,
    ! [M: heap_T5738788834812785303t_unit,Tb: nat,H: heap_e7401611519738050253t_unit] :
      ( aa_nat_o(time_T7469515765551943773t_unit(M),Tb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_t4224138285095624986t_unit(M,H)),Tb) ) ).

% TBOUNDD
tff(fact_4361_TBOUNDD,axiom,
    ! [M: heap_T2636463487746394924on_nat,Tb: nat,H: heap_e7401611519738050253t_unit] :
      ( aa_nat_o(time_T8353473612707095248on_nat(M),Tb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_option_nat(M,H)),Tb) ) ).

% TBOUNDD
tff(fact_4362_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_nat,Tb: nat,H: heap_e7401611519738050253t_unit] :
      ( aa_nat_o(time_TBOUND_nat(M),Tb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_nat(M,H)),Tb) ) ).

% TBOUNDD
tff(fact_4363_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_o,Tb: nat,H: heap_e7401611519738050253t_unit] :
      ( aa_nat_o(time_TBOUND_o(M),Tb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_o(M,H)),Tb) ) ).

% TBOUNDD
tff(fact_4364_TBOUNDD,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,Tb: nat,H: heap_e7401611519738050253t_unit] :
      ( aa_nat_o(time_T5737551269749752165_VEBTi(M),Tb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(M,H)),Tb) ) ).

% TBOUNDD
tff(fact_4365_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_4366_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_4367_TBOUND__prod__case,axiom,
    ! [Tb: product_prod_num_num,Fa: fun_nu4310138114886703706ap_nat,Bnd: fun_num_fun_num_nat] :
      ( ! [A: num,B: num] :
          ( ( Tb = aa_num3156681643174609981um_num(product_Pair_num_num(A),B) )
         => aa_nat_o(time_TBOUND_nat(aa_num1241430908778062596ap_nat(aa_num6639155237539469379ap_nat(Fa,A),B)),aa_num_nat(aa_num_fun_num_nat(Bnd,A),B)) )
     => aa_nat_o(time_TBOUND_nat(produc3422757380792381434ap_nat(Fa,Tb)),produc2914010905598588082um_nat(Bnd,Tb)) ) ).

% TBOUND_prod_case
tff(fact_4368_TBOUND__prod__case,axiom,
    ! [Tb: product_prod_nat_num,Fa: fun_na2425233721303265572ap_nat,Bnd: fun_nat_fun_num_nat] :
      ( ! [A: nat,B: num] :
          ( ( Tb = aa_num5026351239867140595at_num(product_Pair_nat_num(A),B) )
         => aa_nat_o(time_TBOUND_nat(aa_num1241430908778062596ap_nat(aa_nat4754250843956031245ap_nat(Fa,A),B)),aa_num_nat(aa_nat_fun_num_nat(Bnd,A),B)) )
     => aa_nat_o(time_TBOUND_nat(produc8139307783169301188ap_nat(Fa,Tb)),produc4973203077627929192um_nat(Bnd,Tb)) ) ).

% TBOUND_prod_case
tff(fact_4369_TBOUND__prod__case,axiom,
    ! [Tb: product_prod_nat_nat,Fa: fun_na5568033413464243950ap_nat,Bnd: fun_nat_fun_nat_nat] :
      ( ! [A: nat,B: nat] :
          ( ( Tb = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B) )
         => aa_nat_o(time_TBOUND_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(Fa,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(Bnd,A),B)) )
     => aa_nat_o(time_TBOUND_nat(aa_Pro1370750447752133975ap_nat(produc3495815207102967182ap_nat(Fa),Tb)),aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(Bnd),Tb)) ) ).

% TBOUND_prod_case
tff(fact_4370_TBOUND__prod__case,axiom,
    ! [Tb: product_prod_int_int,Fa: fun_in5736188949698081334ap_nat,Bnd: fun_int_fun_int_nat] :
      ( ! [A: int,B: int] :
          ( ( Tb = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) )
         => aa_nat_o(time_TBOUND_nat(aa_int6058887511217079410ap_nat(aa_int2108372073293383455ap_nat(Fa,A),B)),aa_int_nat(aa_int_fun_int_nat(Bnd,A),B)) )
     => aa_nat_o(time_TBOUND_nat(produc2561121167688006358ap_nat(Fa,Tb)),aa_Pro6801748904974264397nt_nat(produc8213879946458358998nt_nat(Bnd),Tb)) ) ).

% TBOUND_prod_case
tff(fact_4371_TBOUND__prod__case,axiom,
    ! [Tb: product_prod_num_num,Fa: fun_nu2951172507782562256Heap_o,Bnd: fun_num_fun_num_nat] :
      ( ! [A: num,B: num] :
          ( ( Tb = aa_num3156681643174609981um_num(product_Pair_num_num(A),B) )
         => aa_nat_o(time_TBOUND_o(aa_num471246959656125306Heap_o(aa_num8283934578203562385Heap_o(Fa,A),B)),aa_num_nat(aa_num_fun_num_nat(Bnd,A),B)) )
     => aa_nat_o(time_TBOUND_o(produc2098883955491630340Heap_o(Fa,Tb)),produc2914010905598588082um_nat(Bnd,Tb)) ) ).

% TBOUND_prod_case
tff(fact_4372_TBOUND__prod__case,axiom,
    ! [Tb: product_prod_nat_num,Fa: fun_na7044599948130059398Heap_o,Bnd: fun_nat_fun_num_nat] :
      ( ! [A: nat,B: num] :
          ( ( Tb = aa_num5026351239867140595at_num(product_Pair_nat_num(A),B) )
         => aa_nat_o(time_TBOUND_o(aa_num471246959656125306Heap_o(aa_nat3153989981696283719Heap_o(Fa,A),B)),aa_num_nat(aa_nat_fun_num_nat(Bnd,A),B)) )
     => aa_nat_o(time_TBOUND_o(produc1066491148394503098Heap_o(Fa,Tb)),produc4973203077627929192um_nat(Bnd,Tb)) ) ).

% TBOUND_prod_case
tff(fact_4373_TBOUND__prod__case,axiom,
    ! [Tb: product_prod_nat_nat,Fa: fun_na1260656955978387516Heap_o,Bnd: fun_nat_fun_nat_nat] :
      ( ! [A: nat,B: nat] :
          ( ( Tb = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B) )
         => aa_nat_o(time_TBOUND_o(aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(Fa,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(Bnd,A),B)) )
     => aa_nat_o(time_TBOUND_o(aa_Pro691548381575413351Heap_o(produc3505292621261808240Heap_o(Fa),Tb)),aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(Bnd),Tb)) ) ).

% TBOUND_prod_case
tff(fact_4374_TBOUND__prod__case,axiom,
    ! [Tb: product_prod_int_int,Fa: fun_in6241394360534301428Heap_o,Bnd: fun_int_fun_int_nat] :
      ( ! [A: int,B: int] :
          ( ( Tb = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) )
         => aa_nat_o(time_TBOUND_o(aa_int6861493060939139724Heap_o(aa_int2288294535011972021Heap_o(Fa,A),B)),aa_int_nat(aa_int_fun_int_nat(Bnd,A),B)) )
     => aa_nat_o(time_TBOUND_o(produc2625828211853636392Heap_o(Fa,Tb)),aa_Pro6801748904974264397nt_nat(produc8213879946458358998nt_nat(Bnd),Tb)) ) ).

% TBOUND_prod_case
tff(fact_4375_TBOUND__prod__case,axiom,
    ! [Tb: product_prod_num_num,Fa: fun_nu2595409035660152255_VEBTi,Bnd: fun_num_fun_num_nat] :
      ( ! [A: num,B: num] :
          ( ( Tb = aa_num3156681643174609981um_num(product_Pair_num_num(A),B) )
         => aa_nat_o(time_T5737551269749752165_VEBTi(aa_num3548024186644552553_VEBTi(aa_num6026116742943590952_VEBTi(Fa,A),B)),aa_num_nat(aa_num_fun_num_nat(Bnd,A),B)) )
     => aa_nat_o(time_T5737551269749752165_VEBTi(produc1633297657014491743_VEBTi(Fa,Tb)),produc2914010905598588082um_nat(Bnd,Tb)) ) ).

% TBOUND_prod_case
tff(fact_4376_TBOUND__prod__case,axiom,
    ! [Tb: product_prod_nat_num,Fa: fun_na8214625747537785481_VEBTi,Bnd: fun_nat_fun_num_nat] :
      ( ! [A: nat,B: num] :
          ( ( Tb = aa_num5026351239867140595at_num(product_Pair_nat_num(A),B) )
         => aa_nat_o(time_T5737551269749752165_VEBTi(aa_num3548024186644552553_VEBTi(aa_nat2421961417966448370_VEBTi(Fa,A),B)),aa_num_nat(aa_nat_fun_num_nat(Bnd,A),B)) )
     => aa_nat_o(time_T5737551269749752165_VEBTi(produc8100297318048944937_VEBTi(Fa,Tb)),produc4973203077627929192um_nat(Bnd,Tb)) ) ).

% TBOUND_prod_case
tff(fact_4377_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,N)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_4378_TBOUND__assert_H__weak,axiom,
    ! [P: $o,M: heap_T2636463487746394924on_nat,Tb: nat] :
      ( ( (P)
       => aa_nat_o(time_T8353473612707095248on_nat(M),Tb) )
     => aa_nat_o(time_T8353473612707095248on_nat(heap_T5999496708990702694on_nat(refine_Imp_assert((P)),aTP_Lamm_km(M))),Tb) ) ).

% TBOUND_assert'_weak
tff(fact_4379_TBOUND__assert_H__weak,axiom,
    ! [P: $o,M: heap_Time_Heap_nat,Tb: nat] :
      ( ( (P)
       => aa_nat_o(time_TBOUND_nat(M),Tb) )
     => aa_nat_o(time_TBOUND_nat(heap_T3781436268274291734it_nat(refine_Imp_assert((P)),aTP_Lamm_kn(M))),Tb) ) ).

% TBOUND_assert'_weak
tff(fact_4380_TBOUND__assert_H__weak,axiom,
    ! [P: $o,M: heap_Time_Heap_o,Tb: nat] :
      ( ( (P)
       => aa_nat_o(time_TBOUND_o(M),Tb) )
     => aa_nat_o(time_TBOUND_o(heap_T7577940988442287570unit_o(refine_Imp_assert((P)),aTP_Lamm_ko(M))),Tb) ) ).

% TBOUND_assert'_weak
tff(fact_4381_TBOUND__assert_H__weak,axiom,
    ! [P: $o,M: heap_T8145700208782473153_VEBTi,Tb: nat] :
      ( ( (P)
       => aa_nat_o(time_T5737551269749752165_VEBTi(M),Tb) )
     => aa_nat_o(time_T5737551269749752165_VEBTi(heap_T7982501707604696571_VEBTi(refine_Imp_assert((P)),aTP_Lamm_kp(M))),Tb) ) ).

% TBOUND_assert'_weak
tff(fact_4382_pred__bound__height,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Tb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))) ) ).

% pred_bound_height
tff(fact_4383_insert__bound__height,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Tb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))))) ) ).

% insert_bound_height
tff(fact_4384_succ__bound__height,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Tb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))) ) ).

% succ_bound_height
tff(fact_4385_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Xb),Xaa) = Y )
     => ( ( ? [A: $o,B: $o] : Xb = vEBT_Leaf((A),(B))
         => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2)
           => ( Y != one_one_nat ) )
         => ( ( ? [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary)
               => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                   => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),
                          $let(
                            xn: nat,
                            xn:= 
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,Xaa),
                            $let(
                              h: nat,
                              h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(
                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                & ~ ( ( Xaa = Mi2 )
                                    | ( Xaa = Ma2 ) ) ),
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(TreeList2,h)))),
                                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Summary),h),one_one_nat)),
                                one_one_nat ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_4386_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,Summary2: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2)),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),
        $let(
          xn: nat,
          xn:= 
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi),Mi,Xb),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
              & ~ ( ( Xb = Mi )
                  | ( Xb = Ma ) ) ),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(TreeList,h)))),
                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Summary2),h),one_one_nat)),
              one_one_nat ) ) )) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_4387_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Xb),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),Xaa)) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2) )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2)),Xaa)) ) )
           => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2)),Xaa)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary) )
                   => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),
                              $let(
                                xn: nat,
                                xn:= 
                                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),Mi2,Xaa),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                    & ~ ( ( Xaa = Mi2 )
                                        | ( Xaa = Ma2 ) ) ),
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,nth_VEBT_VEBT(TreeList2,h)))),
                                      $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_i_n_s_e_r_t,Summary),h),one_one_nat)),
                                    one_one_nat ) ) )) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_4388_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Xb),Xaa) = Y )
     => ( ( ? [Uu2: $o,B: $o] : Xb = vEBT_Leaf((Uu2),(B))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : Xb = vEBT_Leaf((Uv),(Uw))
           => ( ? [N3: nat] : Xaa = aa_nat_nat(suc,N3)
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(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] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Y != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                              one_one_nat,
                              $let(
                                l: nat,
                                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList2,h)))),
                                        $let(
                                          maxlow: option_nat,
                                          maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                            $ite(
                                              ( ( maxlow != none_nat )
                                              & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,nth_VEBT_VEBT(TreeList2,h)),l)),
                                              $let(
                                                sc: option_nat,
                                                sc:= aa_nat_option_nat(vEBT_vebt_succ(Summary),h),
                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Summary),h))),one_one_nat)),
                                                  $ite(sc = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,the_nat(sc)))))) ) )) )),
                                      one_one_nat )) ) ) )) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_4389_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Xb),Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv: $o] : Xb = vEBT_Leaf((Uu2),(Uv))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A: $o,Uw: $o] : Xb = vEBT_Leaf((A),(Uw))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
         => ( ! [A: $o,B: $o] :
                ( ( Xb = vEBT_Leaf((A),(B)) )
               => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
                 => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $ite((B),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(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] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                one_one_nat,
                                $let(
                                  l: nat,
                                  l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
                                            $ite(
                                              ( ( minlow != none_nat )
                                              & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,nth_VEBT_VEBT(TreeList2,h)),l)),
                                              $let(
                                                pr: option_nat,
                                                pr:= aa_nat_option_nat(vEBT_vebt_pred(Summary),h),
                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Summary),h))),one_one_nat)),
                                                  $ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList2,the_nat(pr)))))) ) )) ),
                                        one_one_nat )) ) ) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_4390_log__ceil__idem,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),Xb)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb)))) ) ) ).

% log_ceil_idem
tff(fact_4391_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,Summary2: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2)),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi),
          one_one_nat,
          $let(
            l: nat,
            l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $let(
              h: nat,
              h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList,h)))),
                    $let(
                      maxlow: option_nat,
                      maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                        $ite(
                          ( ( maxlow != none_nat )
                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,nth_VEBT_VEBT(TreeList,h)),l)),
                          $let(
                            sc: option_nat,
                            sc:= aa_nat_option_nat(vEBT_vebt_succ(Summary2),h),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Summary2),h))),one_one_nat)),
                              $ite(sc = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,the_nat(sc)))))) ) )) )),
                  one_one_nat )) ) ) )) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_4392_highsimp,axiom,
    ! [Xb: nat,N: nat] : aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),N)) = aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Xb),N) ).

% highsimp
tff(fact_4393_lowsimp,axiom,
    ! [Xb: nat,N: nat] : aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),N)) = aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Xb),N) ).

% lowsimp
tff(fact_4394_highi__def,axiom,
    ! [Xb: nat,N: nat] : aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Xb),N) = aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% highi_def
tff(fact_4395_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = zero_zero_int )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_4396_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = zero_zero_real )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_4397_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = zero_zero_rat )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_4398_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int = aa_int_int(ring_1_of_int_int,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_4399_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_real = aa_int_real(ring_1_of_int_real,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_4400_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_rat = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_4401_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_4402_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_4403_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_4404_of__int__0,axiom,
    ring_17408606157368542149l_num1(zero_zero_int) = zero_z3563351764282998399l_num1 ).

% of_int_0
tff(fact_4405_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = numeral_numeral_real(N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4406_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = numeral_numeral_rat(N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4407_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,N) )
    <=> ( Z = aa_num_int(numeral_numeral_int,N) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4408_of__int__numeral,axiom,
    ! [K: num] : ring_17408606157368542149l_num1(aa_num_int(numeral_numeral_int,K)) = numera7442385471795722001l_num1(K) ).

% of_int_numeral
tff(fact_4409_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_real(K) ).

% of_int_numeral
tff(fact_4410_of__int__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = numeral_numeral_rat(K) ).

% of_int_numeral
tff(fact_4411_of__int__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,K) ).

% of_int_numeral
tff(fact_4412_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_4413_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_4414_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_4415_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_4416_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_4417_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_4418_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = one_one_int )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4419_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = one_one_real )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4420_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = one_one_rat )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4421_of__int__1,axiom,
    aa_int_uint32(ring_1_of_int_uint32,one_one_int) = one_one_uint32 ).

% of_int_1
tff(fact_4422_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_4423_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_4424_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_4425_of__int__1,axiom,
    ring_17408606157368542149l_num1(one_one_int) = one_on7727431528512463931l_num1 ).

% of_int_1
tff(fact_4426_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Xb) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4427_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Xb) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4428_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Xb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4429_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Xb) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_int_complex(ring_17405671764205052669omplex,B2)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4430_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Xb) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B2)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4431_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W) = aa_int_rat(ring_1_of_int_rat,Xb) )
    <=> ( aa_nat_int(power_power_int(B2),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4432_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W) = aa_int_int(ring_1_of_int_int,Xb) )
    <=> ( aa_nat_int(power_power_int(B2),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4433_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W) = aa_int_real(ring_1_of_int_real,Xb) )
    <=> ( aa_nat_int(power_power_int(B2),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4434_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_int_complex(ring_17405671764205052669omplex,B2)),W) = aa_int_complex(ring_17405671764205052669omplex,Xb) )
    <=> ( aa_nat_int(power_power_int(B2),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4435_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B2)),W) = aa_int_Code_integer(ring_18347121197199848620nteger,Xb) )
    <=> ( aa_nat_int(power_power_int(B2),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4436_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z),N)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),N) ).

% of_int_power
tff(fact_4437_of__int__power,axiom,
    ! [Z: int,N: nat] : ring_17408606157368542149l_num1(aa_nat_int(power_power_int(Z),N)) = aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(ring_17408606157368542149l_num1(Z)),N) ).

% of_int_power
tff(fact_4438_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),N)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),N) ).

% of_int_power
tff(fact_4439_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z)),N) ).

% of_int_power
tff(fact_4440_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z),N)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_int_complex(ring_17405671764205052669omplex,Z)),N) ).

% of_int_power
tff(fact_4441_of__int__power,axiom,
    ! [Z: int,N: nat] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_nat_int(power_power_int(Z),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,Z)),N) ).

% of_int_power
tff(fact_4442_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_4443_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_4444_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_4445_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_4446_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_4447_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_4448_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(N)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_4449_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(N)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_4450_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,N)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_4451_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),numeral_numeral_real(N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N)) ) ).

% of_int_le_numeral_iff
tff(fact_4452_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),numeral_numeral_rat(N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N)) ) ).

% of_int_le_numeral_iff
tff(fact_4453_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,N)) ) ).

% of_int_le_numeral_iff
tff(fact_4454_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_4455_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_4456_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_4457_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_4458_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_4459_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_4460_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(N)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_4461_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(N)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_4462_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,N)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_4463_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),numeral_numeral_real(N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,N)) ) ).

% of_int_less_numeral_iff
tff(fact_4464_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),numeral_numeral_rat(N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,N)) ) ).

% of_int_less_numeral_iff
tff(fact_4465_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,N)) ) ).

% of_int_less_numeral_iff
tff(fact_4466_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_4467_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_4468_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_4469_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_4470_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_4471_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_4472_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_4473_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_4474_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_4475_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_4476_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_4477_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_4478_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,N: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(Xb)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4479_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,N: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xb)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4480_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,N: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4481_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,N: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4482_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4483_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: int] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(Xb)),N) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4484_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xb)),N) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4485_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),N) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4486_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),N) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4487_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4488_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4489_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4490_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Xb)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4491_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Xb)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4492_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),W)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4493_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B2)),W)),aa_int_Code_integer(ring_18347121197199848620nteger,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),W)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4494_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),W)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4495_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),W)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4496_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4497_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4498_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Xb)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4499_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Xb)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4500_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,B2)),W)),aa_int_Code_integer(ring_18347121197199848620nteger,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),W)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4501_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),W)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4502_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),W)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4503_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),W)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4504_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xb)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4505_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4506_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A2)),aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4507_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4508_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xb)),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)),A2) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4509_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),N)),aa_int_real(ring_1_of_int_real,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)),A2) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4510_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),N)),aa_int_rat(ring_1_of_int_rat,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)),A2) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4511_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)),aa_int_int(ring_1_of_int_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)),A2) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4512_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xb)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4513_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4514_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A2)),aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4515_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4516_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(Xb)),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)),A2) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4517_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(Xb)),N)),aa_int_real(ring_1_of_int_real,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)),A2) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4518_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(numeral_numeral_rat(Xb)),N)),aa_int_rat(ring_1_of_int_rat,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)),A2) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4519_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)),aa_int_int(ring_1_of_int_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)),A2) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4520_ex__le__of__int,axiom,
    ! [Xb: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_le_of_int
tff(fact_4521_ex__le__of__int,axiom,
    ! [Xb: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_le_of_int
tff(fact_4522_ex__of__int__less,axiom,
    ! [Xb: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),Xb) ).

% ex_of_int_less
tff(fact_4523_ex__of__int__less,axiom,
    ! [Xb: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),Xb) ).

% ex_of_int_less
tff(fact_4524_ex__less__of__int,axiom,
    ! [Xb: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_less_of_int
tff(fact_4525_ex__less__of__int,axiom,
    ! [Xb: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_less_of_int
tff(fact_4526_le__of__int__ceiling,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))) ).

% le_of_int_ceiling
tff(fact_4527_le__of__int__ceiling,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(Xb))) ).

% le_of_int_ceiling
tff(fact_4528_ceiling__le__iff,axiom,
    ! [Xb: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z)) ) ).

% ceiling_le_iff
tff(fact_4529_ceiling__le__iff,axiom,
    ! [Xb: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xb)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% ceiling_le_iff
tff(fact_4530_ceiling__le,axiom,
    ! [Xb: real,A2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,A2))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),A2) ) ).

% ceiling_le
tff(fact_4531_ceiling__le,axiom,
    ! [Xb: rat,A2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,A2))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xb)),A2) ) ).

% ceiling_le
tff(fact_4532_less__ceiling__iff,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xb) ) ).

% less_ceiling_iff
tff(fact_4533_less__ceiling__iff,axiom,
    ! [Z: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),Xb) ) ).

% less_ceiling_iff
tff(fact_4534_real__of__int__div4,axiom,
    ! [N: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),Xb))),aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,Xb))) ).

% real_of_int_div4
tff(fact_4535_minNull__bound,axiom,
    ! [Tb: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Tb)),one_one_nat) ).

% minNull_bound
tff(fact_4536_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,
    aa_VEBT_VEBT_nat(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)
tff(fact_4537_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: $o] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf($true,(Uv2))) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
tff(fact_4538_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,
    ! [Uu: $o] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,vEBT_Leaf((Uu),$true)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
tff(fact_4539_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).

% of_int_nonneg
tff(fact_4540_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_nonneg
tff(fact_4541_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_nonneg
tff(fact_4542_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).

% of_int_pos
tff(fact_4543_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_pos
tff(fact_4544_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_pos
tff(fact_4545_floor__exists1,axiom,
    ! [Xb: real] :
    ? [X2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X2)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),one_one_int)))
      & ! [Y4: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y4)),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int))) )
         => ( Y4 = X2 ) ) ) ).

% floor_exists1
tff(fact_4546_floor__exists1,axiom,
    ! [Xb: rat] :
    ? [X2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X2)),Xb)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),one_one_int)))
      & ! [Y4: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y4)),Xb)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int))) )
         => ( Y4 = X2 ) ) ) ).

% floor_exists1
tff(fact_4547_floor__exists,axiom,
    ! [Xb: real] :
    ? [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int))) ) ).

% floor_exists
tff(fact_4548_floor__exists,axiom,
    ! [Xb: rat] :
    ? [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),Xb)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int))) ) ).

% floor_exists
tff(fact_4549_of__int__ceiling__le__add__one,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),one_one_real)) ).

% of_int_ceiling_le_add_one
tff(fact_4550_of__int__ceiling__le__add__one,axiom,
    ! [R: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(R))),plus_plus_rat(R,one_one_rat)) ).

% of_int_ceiling_le_add_one
tff(fact_4551_of__int__ceiling__diff__one__le,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R))),one_one_real)),R) ).

% of_int_ceiling_diff_one_le
tff(fact_4552_of__int__ceiling__diff__one__le,axiom,
    ! [R: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(R)),one_one_rat)),R) ).

% of_int_ceiling_diff_one_le
tff(fact_4553_of__nat__less__of__int__iff,axiom,
    ! [N: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,N)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_4554_of__nat__less__of__int__iff,axiom,
    ! [N: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_4555_of__nat__less__of__int__iff,axiom,
    ! [N: nat,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_4556_of__nat__less__of__int__iff,axiom,
    ! [N: nat,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,N)),aa_int_Code_integer(ring_18347121197199848620nteger,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_4557_int__le__real__less,axiom,
    ! [N: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,N),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,N)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,M)),one_one_real)) ) ).

% int_le_real_less
tff(fact_4558_int__less__real__le,axiom,
    ! [N: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,N),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real)),aa_int_real(ring_1_of_int_real,M)) ) ).

% int_less_real_le
tff(fact_4559_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N))
     => ( set_or1266510415728281911st_int(M,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N),set_or1266510415728281911st_int(M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_4560_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,vEBT_Leaf((A2),(B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite((B2),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
tff(fact_4561_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1266510415728281911st_int(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),bot_bot_set_int,insert_int(I,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J))) ).

% simp_from_to
tff(fact_4562_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv2),Uw2)) = one_one_nat ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
tff(fact_4563_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv2),Uw2)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
tff(fact_4564_ceiling__correct,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))),one_one_real)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))) ) ).

% ceiling_correct
tff(fact_4565_ceiling__correct,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(Xb)),one_one_rat)),Xb)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(Xb))) ) ).

% ceiling_correct
tff(fact_4566_ceiling__unique,axiom,
    ! [Z: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z))
       => ( aa_real_int(archim7802044766580827645g_real,Xb) = Z ) ) ) ).

% ceiling_unique
tff(fact_4567_ceiling__unique,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z))
       => ( archim2889992004027027881ng_rat(Xb) = Z ) ) ) ).

% ceiling_unique
tff(fact_4568_ceiling__eq__iff,axiom,
    ! [Xb: real,A2: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,Xb) = A2 )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,A2)),one_one_real)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,A2)) ) ) ).

% ceiling_eq_iff
tff(fact_4569_ceiling__eq__iff,axiom,
    ! [Xb: rat,A2: int] :
      ( ( archim2889992004027027881ng_rat(Xb) = A2 )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A2),one_one_rat)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,A2)) ) ) ).

% ceiling_eq_iff
tff(fact_4570_ceiling__split,axiom,
    ! [P: fun_int_o,Tb: real] :
      ( aa_int_o(P,aa_real_int(archim7802044766580827645g_real,Tb))
    <=> ! [I4: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,I4)),one_one_real)),Tb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Tb),aa_int_real(ring_1_of_int_real,I4)) )
         => aa_int_o(P,I4) ) ) ).

% ceiling_split
tff(fact_4571_ceiling__split,axiom,
    ! [P: fun_int_o,Tb: rat] :
      ( aa_int_o(P,archim2889992004027027881ng_rat(Tb))
    <=> ! [I4: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I4),one_one_rat)),Tb)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Tb),aa_int_rat(ring_1_of_int_rat,I4)) )
         => aa_int_o(P,I4) ) ) ).

% ceiling_split
tff(fact_4572_ceiling__less__iff,axiom,
    ! [Xb: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% ceiling_less_iff
tff(fact_4573_ceiling__less__iff,axiom,
    ! [Xb: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xb)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)) ) ).

% ceiling_less_iff
tff(fact_4574_le__ceiling__iff,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z),one_one_rat)),Xb) ) ).

% le_ceiling_iff
tff(fact_4575_le__ceiling__iff,axiom,
    ! [Z: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xb) ) ).

% le_ceiling_iff
tff(fact_4576_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] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_VEB5832953974749077948T_VEBT(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)
tff(fact_4577_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] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_VEB5832953974749077948T_VEBT(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)
tff(fact_4578_real__of__int__div2,axiom,
    ! [N: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,Xb))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),Xb)))) ).

% real_of_int_div2
tff(fact_4579_real__of__int__div3,axiom,
    ! [N: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,Xb))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),Xb)))),one_one_real) ).

% real_of_int_div3
tff(fact_4580_maxt__bound,axiom,
    ! [Tb: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Tb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ).

% maxt_bound
tff(fact_4581_ceiling__divide__upper,axiom,
    ! [Q2: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,P2),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(P2),Q2)))),Q2)) ) ).

% ceiling_divide_upper
tff(fact_4582_ceiling__divide__upper,axiom,
    ! [Q2: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P2,Q2)))),Q2)) ) ).

% ceiling_divide_upper
tff(fact_4583_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] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux,Uy),Uz)) = one_one_nat ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_4584_mint__bound,axiom,
    ! [Tb: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Tb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ).

% mint_bound
tff(fact_4585_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,vEBT_Leaf((A2),(B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite((A2),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
tff(fact_4586_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] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux,Uy),Uz)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_4587_ceiling__divide__lower,axiom,
    ! [Q2: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(P2),Q2)))),one_one_real)),Q2)),P2) ) ).

% ceiling_divide_lower
tff(fact_4588_ceiling__divide__lower,axiom,
    ! [Q2: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim2889992004027027881ng_rat(divide_divide_rat(P2,Q2))),one_one_rat)),Q2)),P2) ) ).

% ceiling_divide_lower
tff(fact_4589_ceiling__eq,axiom,
    ! [N: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,N)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real))
       => ( aa_real_int(archim7802044766580827645g_real,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_4590_ceiling__eq,axiom,
    ! [N: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,N)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),plus_plus_rat(aa_int_rat(ring_1_of_int_rat,N),one_one_rat))
       => ( archim2889992004027027881ng_rat(Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_4591_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,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Xb) = Y )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uv: $o] : Xb = vEBT_Leaf($true,(Uv))
           => ( Y != one_one_nat ) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => ( Y != one_one_nat ) )
           => ( ( ? [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2),Uy2)
               => ( Y != one_one_nat ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2)
                 => ( Y != one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_4592_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Xb) = Y )
     => ( ! [A: $o,B: $o] :
            ( ( Xb = vEBT_Leaf((A),(B)) )
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                  $ite((B),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2)
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_4593_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Xb) = Y )
     => ( ! [A: $o] :
            ( ? [B: $o] : Xb = vEBT_Leaf((A),(B))
           => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                  $ite((A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2)
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_4594_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,Summary2: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2)),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
        $ite(
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xb) ),
          one_one_nat,
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
            $ite(
              ( ( Xb = Mi )
              & ( Xb = Ma ) ),
              aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),
                        $ite(Xb = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary2)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),one_one_nat))),one_one_nat)),
                $let(
                  xn: nat,
                  xn:= 
                    $ite(Xb = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary2)))))),Xb),
                  $let(
                    l: nat,
                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $let(
                      h: nat,
                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,nth_VEBT_VEBT(TreeList,h)),l))),
                          $let(
                            newnode: vEBT_VEBT,
                            newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,h),l),
                            $let(
                              newlist: list_VEBT_VEBT,
                              newlist:= list_u1324408373059187874T_VEBT(TreeList,h,newnode),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,newnode))),
                                $ite(
                                  aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Summary2),h))),
                                    $let(
                                      sn: vEBT_VEBT,
                                      sn:= vEBT_vebt_delete(Summary2,h),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                        $ite(
                                          xn = Ma,
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,sn))),
                                            $let(
                                              maxs: option_nat,
                                              maxs:= vEBT_vebt_maxt(sn),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                                $ite(maxs = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,the_nat(maxs)))))) )),
                                          one_one_nat )) )),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                    $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,h))),one_one_nat)) )) ) )),
                        one_one_nat ) ) ) )) )) )) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
tff(fact_4595_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Xb),Xaa) = Y )
     => ( ( ? [A: $o,B: $o] : Xb = vEBT_Leaf((A),(B))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A: $o,B: $o] : Xb = vEBT_Leaf((A),(B))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A: $o,B: $o] : Xb = vEBT_Leaf((A),(B))
             => ( ? [N3: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N3))
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2),Summary)
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,TreeList2),Summary)
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary)
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                              $ite(
                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                                | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                                one_one_nat,
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)),
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),
                                              $ite(Xaa = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),one_one_nat))),one_one_nat)),
                                      $let(
                                        xn: nat,
                                        xn:= 
                                          $ite(Xaa = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                        $let(
                                          l: nat,
                                          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $let(
                                            h: nat,
                                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                            $ite(
                                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,nth_VEBT_VEBT(TreeList2,h)),l))),
                                                $let(
                                                  newnode: vEBT_VEBT,
                                                  newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),l),
                                                  $let(
                                                    newlist: list_VEBT_VEBT,
                                                    newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode),
                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,newnode))),
                                                      $ite(
                                                        aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Summary),h))),
                                                          $let(
                                                            sn: vEBT_VEBT,
                                                            sn:= vEBT_vebt_delete(Summary,h),
                                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                                              $ite(
                                                                xn = Ma2,
                                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,sn))),
                                                                  $let(
                                                                    maxs: option_nat,
                                                                    maxs:= vEBT_vebt_maxt(sn),
                                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                                                      $ite(maxs = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,the_nat(maxs)))))) )),
                                                                one_one_nat )) )),
                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                                          $ite(xn = Ma2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,h))),one_one_nat)) )) ) )),
                                              one_one_nat ) ) ) )) )) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
tff(fact_4596_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,Summary2: vEBT_VEBT,Xb: nat] :
      aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList),Summary2)),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xb),
          one_one_nat,
          $let(
            l: nat,
            l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $let(
              h: nat,
              h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList,h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
                      $ite(
                        ( ( minlow != none_nat )
                        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,nth_VEBT_VEBT(TreeList,h)),l)),
                        $let(
                          pr: option_nat,
                          pr:= aa_nat_option_nat(vEBT_vebt_pred(Summary2),h),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Summary2),h))),one_one_nat)),
                            $ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList,the_nat(pr)))))) ) )) ),
                  one_one_nat )) ) ) )) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_4597_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Xb),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),zero_zero_nat)) ) ) )
         => ( ! [A: $o,B: $o] :
                ( ( Xb = vEBT_Leaf((A),(B)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A: $o,B: $o] :
                  ( ( Xb = vEBT_Leaf((A),(B)) )
                 => ! [N3: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,N3)))) ) ) )
             => ( ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2),Summary) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList2),Summary)),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,TreeList2),Summary) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),zero_zero_nat,TreeList2),Summary)),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                         => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2)
                                    | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa) ),
                                    one_one_nat,
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                      $ite(
                                        ( ( Xaa = Mi2 )
                                        & ( Xaa = Ma2 ) ),
                                        aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),
                                                  $ite(Xaa = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),one_one_nat))),one_one_nat)),
                                          $let(
                                            xn: nat,
                                            xn:= 
                                              $ite(Xaa = Mi2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                            $let(
                                              l: nat,
                                              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                              $let(
                                                h: nat,
                                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                                $ite(
                                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,nth_VEBT_VEBT(TreeList2,h)),l))),
                                                    $let(
                                                      newnode: vEBT_VEBT,
                                                      newnode:= vEBT_vebt_delete(nth_VEBT_VEBT(TreeList2,h),l),
                                                      $let(
                                                        newlist: list_VEBT_VEBT,
                                                        newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode),
                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,newnode))),
                                                          $ite(
                                                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode),
                                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_d_e_l_e_t_e,Summary),h))),
                                                              $let(
                                                                sn: vEBT_VEBT,
                                                                sn:= vEBT_vebt_delete(Summary,h),
                                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                                                  $ite(
                                                                    xn = Ma2,
                                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,sn))),
                                                                      $let(
                                                                        maxs: option_nat,
                                                                        maxs:= vEBT_vebt_maxt(sn),
                                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                                                          $ite(maxs = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,the_nat(maxs)))))) )),
                                                                    one_one_nat )) )),
                                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                                              $ite(xn = Ma2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(newlist,h))),one_one_nat)) )) ) )),
                                                  one_one_nat ) ) ) )) )) )) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_4598_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Xb),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(B))),zero_zero_nat)) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( Xb = vEBT_Leaf((Uv),(Uw)) )
               => ! [N3: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N3) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw))),aa_nat_nat(suc,N3))) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(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,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc2),Vd2)),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Y = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi2),
                                  one_one_nat,
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),
                                        $ite(
                                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList2,h)))),
                                            $let(
                                              maxlow: option_nat,
                                              maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,h)),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                                $ite(
                                                  ( ( maxlow != none_nat )
                                                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,l)),maxlow) ),
                                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,nth_VEBT_VEBT(TreeList2,h)),l)),
                                                  $let(
                                                    sc: option_nat,
                                                    sc:= aa_nat_option_nat(vEBT_vebt_succ(Summary),h),
                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_s_u_c_c,Summary),h))),one_one_nat)),
                                                      $ite(sc = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,the_nat(sc)))))) ) )) )),
                                          one_one_nat )) ) ) )) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_4599_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Xb),Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv))),zero_zero_nat)) ) ) )
         => ( ! [A: $o,Uw: $o] :
                ( ( Xb = vEBT_Leaf((A),(Uw)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(Uw))),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A: $o,B: $o] :
                  ( ( Xb = vEBT_Leaf((A),(B)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite((B),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((A),(B))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3) )
                   => ( ( Y = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Va3)),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(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,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2),Ve2)),Xaa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Y = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                         => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xaa),
                                    one_one_nat,
                                    $let(
                                      l: nat,
                                      l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $let(
                                        h: nat,
                                        h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                            $let(
                                              minlow: option_nat,
                                              minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,nth_VEBT_VEBT(TreeList2,h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
                                                $ite(
                                                  ( ( minlow != none_nat )
                                                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,l)),minlow) ),
                                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,nth_VEBT_VEBT(TreeList2,h)),l)),
                                                  $let(
                                                    pr: option_nat,
                                                    pr:= aa_nat_option_nat(vEBT_vebt_pred(Summary),h),
                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_VEB1122092716774696213at_nat(vEBT_T_p_r_e_d,Summary),h))),one_one_nat)),
                                                      $ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,nth_VEBT_VEBT(TreeList2,the_nat(pr)))))) ) )) ),
                                            one_one_nat )) ) ) )) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_4600_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_lu),unique5055182867167087721od_nat(M,N)) ).

% divmod_algorithm_code(6)
tff(fact_4601_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,N)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_lv),unique5052692396658037445od_int(M,N)) ).

% divmod_algorithm_code(6)
tff(fact_4602_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K5: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N))),H)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_4603_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K5: real,N: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Z),H)),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),N))),H)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(semiri8010041392384452111omplex,N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_4604_foldr__zero,axiom,
    ! [Xs: list_nat,D: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(nth_nat(Xs),I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),D)),D)) ) ).

% foldr_zero
tff(fact_4605_foldr__one,axiom,
    ! [D: nat,Ys: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D)) ).

% foldr_one
tff(fact_4606_foldr__same__int,axiom,
    ! [Xs: list_nat,Y: nat] :
      ( ! [X2: nat,Y2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => ( aa_set_nat_o(member_nat(Y2),set_nat2(Xs))
           => ( X2 = Y2 ) ) )
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
           => ( X2 = Y ) )
       => ( aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),Y) ) ) ) ).

% foldr_same_int
tff(fact_4607_foldr__mono,axiom,
    ! [Xs: list_nat,Ys: list_nat,C: nat,D: nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(nth_nat(Xs),I2)),aa_nat_nat(nth_nat(Ys),I2)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),C)),size_size_list_nat(Ys))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D)) ) ) ) ).

% foldr_mono
tff(fact_4608_word__of__int__numeral,axiom,
    ! [Bin: num] : ring_17408606157368542149l_num1(aa_num_int(numeral_numeral_int,Bin)) = numera7442385471795722001l_num1(Bin) ).

% word_of_int_numeral
tff(fact_4609_word__of__int__0,axiom,
    ring_17408606157368542149l_num1(zero_zero_int) = zero_z3563351764282998399l_num1 ).

% word_of_int_0
tff(fact_4610_word__of__int__1,axiom,
    ring_17408606157368542149l_num1(one_one_int) = one_on7727431528512463931l_num1 ).

% word_of_int_1
tff(fact_4611_numeral__div__numeral,axiom,
    ! [K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(K,L)) ).

% numeral_div_numeral
tff(fact_4612_numeral__div__numeral,axiom,
    ! [K: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,L)) = aa_Pro6799258434465214121nt_int(product_fst_int_int,unique5052692396658037445od_int(K,L)) ).

% numeral_div_numeral
tff(fact_4613_foldr__length,axiom,
    ! [L: list_real] : aa_nat_nat(foldr_real_nat(aTP_Lamm_lw,L),zero_zero_nat) = size_size_list_real(L) ).

% foldr_length
tff(fact_4614_foldr__length,axiom,
    ! [L: list_o] : aa_nat_nat(foldr_o_nat(aTP_Lamm_lx,L),zero_zero_nat) = size_size_list_o(L) ).

% foldr_length
tff(fact_4615_foldr__length,axiom,
    ! [L: list_nat] : aa_nat_nat(foldr_nat_nat(aTP_Lamm_ly,L),zero_zero_nat) = size_size_list_nat(L) ).

% foldr_length
tff(fact_4616_foldr__length,axiom,
    ! [L: list_int] : aa_nat_nat(foldr_int_nat(aTP_Lamm_lz,L),zero_zero_nat) = size_size_list_int(L) ).

% foldr_length
tff(fact_4617_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5055182867167087721od_nat(M,one) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_num_nat(numeral_numeral_nat,M)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_4618_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5052692396658037445od_int(M,one) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,M)),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_4619_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,N)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_4620_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,N)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_4621_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,N)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_4622_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,N)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_4623_Suc__0__div__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_div_numeral
tff(fact_4624_one__div__numeral,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,N)) ).

% one_div_numeral
tff(fact_4625_one__div__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,N)) = aa_Pro6799258434465214121nt_int(product_fst_int_int,unique5052692396658037445od_int(one,N)) ).

% one_div_numeral
tff(fact_4626_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,N)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ma),unique5055182867167087721od_nat(M,N)) ).

% divmod_algorithm_code(5)
tff(fact_4627_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,N)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_mb),unique5052692396658037445od_int(M,N)) ).

% divmod_algorithm_code(5)
tff(fact_4628_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,N)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,N)),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ).

% divmod_algorithm_code(7)
tff(fact_4629_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,N)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(aa_num_num(bit1,N)),unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ).

% divmod_algorithm_code(7)
tff(fact_4630_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,N)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),N),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,M))),unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ).

% divmod_algorithm_code(7)
tff(fact_4631_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,N)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,N)),unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ).

% divmod_algorithm_code(8)
tff(fact_4632_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,N)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(aa_num_num(bit1,N)),unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ).

% divmod_algorithm_code(8)
tff(fact_4633_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,N)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,M))),unique4921790084139445826nteger(aa_num_num(bit1,N),unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,N))))) ).

% divmod_algorithm_code(8)
tff(fact_4634_word__numeral__alt,axiom,
    ! [B2: num] : numera7442385471795722001l_num1(B2) = ring_17408606157368542149l_num1(aa_num_int(numeral_numeral_int,B2)) ).

% word_numeral_alt
tff(fact_4635_word__of__int__power__hom,axiom,
    ! [A2: int,N: nat] : aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(ring_17408606157368542149l_num1(A2)),N) = ring_17408606157368542149l_num1(aa_nat_int(power_power_int(A2),N)) ).

% word_of_int_power_hom
tff(fact_4636_foldr__length__aux,axiom,
    ! [L: list_real,A2: nat] : aa_nat_nat(foldr_real_nat(aTP_Lamm_lw,L),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),size_size_list_real(L)) ).

% foldr_length_aux
tff(fact_4637_foldr__length__aux,axiom,
    ! [L: list_o,A2: nat] : aa_nat_nat(foldr_o_nat(aTP_Lamm_lx,L),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),size_size_list_o(L)) ).

% foldr_length_aux
tff(fact_4638_foldr__length__aux,axiom,
    ! [L: list_nat,A2: nat] : aa_nat_nat(foldr_nat_nat(aTP_Lamm_ly,L),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),size_size_list_nat(L)) ).

% foldr_length_aux
tff(fact_4639_foldr__length__aux,axiom,
    ! [L: list_int,A2: nat] : aa_nat_nat(foldr_int_nat(aTP_Lamm_lz,L),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),size_size_list_int(L)) ).

% foldr_length_aux
tff(fact_4640_fst__divmod,axiom,
    ! [M: num,N: num] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) ).

% fst_divmod
tff(fact_4641_fst__divmod,axiom,
    ! [M: num,N: num] : aa_Pro6799258434465214121nt_int(product_fst_int_int,unique5052692396658037445od_int(M,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) ).

% fst_divmod
tff(fact_4642_word__of__int__2p,axiom,
    ! [N: nat] : ring_17408606157368542149l_num1(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N) ).

% word_of_int_2p
tff(fact_4643_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      unique5052692396658037445od_int(M,N) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,M)),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(N),unique5052692396658037445od_int(M,aa_num_num(bit0,N)))) ).

% divmod_divmod_step
tff(fact_4644_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      unique5055182867167087721od_nat(M,N) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,M)),aa_Pro262675715343573244at_nat(unique5026877609467782581ep_nat(N),unique5055182867167087721od_nat(M,aa_num_num(bit0,N)))) ).

% divmod_divmod_step
tff(fact_4645_divmod__divmod__step,axiom,
    ! [M: num,N: num] :
      unique3479559517661332726nteger(M,N) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),N),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(M)),unique4921790084139445826nteger(N,unique3479559517661332726nteger(M,aa_num_num(bit0,N)))) ).

% divmod_divmod_step
tff(fact_4646_rel__of__def,axiom,
    ! [M: fun_na5542100196373022448T_VEBT,P: fun_Pr6320724876807202603VEBT_o] : rel_of7001861737123029207T_VEBT(M,P) = collec1047362574656026267T_VEBT(produc2834603712688810931VEBT_o(aa_fun1244858531897282742VEBT_o(aTP_Lamm_mc(M),P))) ).

% rel_of_def
tff(fact_4647_rel__of__def,axiom,
    ! [M: fun_num_option_num,P: fun_Pr6331313853612757382_num_o] : rel_of_num_num(M,P) = collec2230928802738392704um_num(produc5703948589228662326_num_o(aa_fun1566934675525854290_num_o(aTP_Lamm_md(M),P))) ).

% rel_of_def
tff(fact_4648_rel__of__def,axiom,
    ! [M: fun_nat_option_num,P: fun_Pr5555124106300582480_num_o] : rel_of_nat_num(M,P) = collec4100598399430923318at_num(produc4927758841916487424_num_o(aa_fun8093532814315391442_num_o(aTP_Lamm_me(M),P))) ).

% rel_of_def
tff(fact_4649_rel__of__def,axiom,
    ! [M: fun_nat_option_nat,P: fun_Pr4796943543331071514_nat_o] : rel_of_nat_nat(M,P) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aa_fun2218514334412473170_nat_o(aTP_Lamm_mf(M),P))) ).

% rel_of_def
tff(fact_4650_rel__of__def,axiom,
    ! [M: fun_int_option_int,P: fun_Pr3238374608701176162_int_o] : rel_of_int_int(M,P) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aa_fun5140975492127840594_int_o(aTP_Lamm_mg(M),P))) ).

% rel_of_def
tff(fact_4651_norm__divide__numeral,axiom,
    ! [A2: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(A2),numeral_numeral_real(W))) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,A2)),numeral_numeral_real(W)) ).

% norm_divide_numeral
tff(fact_4652_norm__divide__numeral,axiom,
    ! [A2: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),numera6690914467698888265omplex(W))) = aa_real_real(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,A2)),numeral_numeral_real(W)) ).

% norm_divide_numeral
tff(fact_4653_norm__mult__numeral1,axiom,
    ! [W: num,A2: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(numeral_numeral_real(W)),A2)) = aa_real_real(times_times_real(numeral_numeral_real(W)),aa_real_real(real_V7735802525324610683m_real,A2)) ).

% norm_mult_numeral1
tff(fact_4654_norm__mult__numeral1,axiom,
    ! [W: num,A2: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),A2)) = aa_real_real(times_times_real(numeral_numeral_real(W)),aa_complex_real(real_V1022390504157884413omplex,A2)) ).

% norm_mult_numeral1
tff(fact_4655_norm__mult__numeral2,axiom,
    ! [A2: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(A2),numeral_numeral_real(W))) = aa_real_real(times_times_real(aa_real_real(real_V7735802525324610683m_real,A2)),numeral_numeral_real(W)) ).

% norm_mult_numeral2
tff(fact_4656_norm__mult__numeral2,axiom,
    ! [A2: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),numera6690914467698888265omplex(W))) = aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,A2)),numeral_numeral_real(W)) ).

% norm_mult_numeral2
tff(fact_4657_norm__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xb)),zero_zero_real)
    <=> ( Xb = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_4658_norm__le__zero__iff,axiom,
    ! [Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),zero_zero_real)
    <=> ( Xb = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_4659_zero__less__norm__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,Xb))
    <=> ( Xb != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_4660_zero__less__norm__iff,axiom,
    ! [Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xb))
    <=> ( Xb != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_4661_norm__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,numeral_numeral_real(W)) = numeral_numeral_real(W) ).

% norm_numeral
tff(fact_4662_norm__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = numeral_numeral_real(W) ).

% norm_numeral
tff(fact_4663_norm__one,axiom,
    aa_real_real(real_V7735802525324610683m_real,one_one_real) = one_one_real ).

% norm_one
tff(fact_4664_norm__one,axiom,
    aa_complex_real(real_V1022390504157884413omplex,one_one_complex) = one_one_real ).

% norm_one
tff(fact_4665_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_4666_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_4667_norm__eq__zero,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(real_V7735802525324610683m_real,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% norm_eq_zero
tff(fact_4668_norm__eq__zero,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_4669_norm__not__less__zero,axiom,
    ! [Xb: complex] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),zero_zero_real) ).

% norm_not_less_zero
tff(fact_4670_norm__ge__zero,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% norm_ge_zero
tff(fact_4671_norm__divide,axiom,
    ! [A2: real,B2: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(A2),B2)) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B2)) ).

% norm_divide
tff(fact_4672_norm__divide,axiom,
    ! [A2: complex,B2: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2)) = aa_real_real(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B2)) ).

% norm_divide
tff(fact_4673_norm__power,axiom,
    ! [Xb: real,N: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xb)),N) ).

% norm_power
tff(fact_4674_norm__power,axiom,
    ! [Xb: complex,N: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),N) ).

% norm_power
tff(fact_4675_power__eq__imp__eq__norm,axiom,
    ! [W: real,N: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),N) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_4676_power__eq__imp__eq__norm,axiom,
    ! [W: complex,N: nat,Z: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),N) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),N) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_complex_real(real_V1022390504157884413omplex,W) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_4677_nonzero__norm__divide,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(A2),B2)) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B2)) ) ) ).

% nonzero_norm_divide
tff(fact_4678_nonzero__norm__divide,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2)) = aa_real_real(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B2)) ) ) ).

% nonzero_norm_divide
tff(fact_4679_norm__mult__less,axiom,
    ! [Xb: real,R: real,Y: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(Xb),Y))),aa_real_real(times_times_real(R),S)) ) ) ).

% norm_mult_less
tff(fact_4680_norm__mult__less,axiom,
    ! [Xb: complex,R: real,Y: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Y))),aa_real_real(times_times_real(R),S)) ) ) ).

% norm_mult_less
tff(fact_4681_norm__triangle__lt,axiom,
    ! [Xb: real,Y: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),E2) ) ).

% norm_triangle_lt
tff(fact_4682_norm__triangle__lt,axiom,
    ! [Xb: complex,Y: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y))),E2) ) ).

% norm_triangle_lt
tff(fact_4683_norm__add__less,axiom,
    ! [Xb: real,R: real,Y: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),S)) ) ) ).

% norm_add_less
tff(fact_4684_norm__add__less,axiom,
    ! [Xb: complex,R: real,Y: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R),S)) ) ) ).

% norm_add_less
tff(fact_4685_norm__diff__triangle__less,axiom,
    ! [Xb: real,Y: real,E1: real,Z: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_4686_norm__diff__triangle__less,axiom,
    ! [Xb: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_4687_norm__power__ineq,axiom,
    ! [Xb: real,N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xb)),N)) ).

% norm_power_ineq
tff(fact_4688_norm__power__ineq,axiom,
    ! [Xb: complex,N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),N)) ).

% norm_power_ineq
tff(fact_4689_power__eq__1__iff,axiom,
    ! [W: real,N: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),N) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_4690_power__eq__1__iff,axiom,
    ! [W: complex,N: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),N) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W) = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_4691_square__norm__one,axiom,
    ! [Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
     => ( aa_real_real(real_V7735802525324610683m_real,Xb) = one_one_real ) ) ).

% square_norm_one
tff(fact_4692_square__norm__one,axiom,
    ! [Xb: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,Xb) = one_one_real ) ) ).

% square_norm_one
tff(fact_4693_norm__power__diff,axiom,
    ! [Z: real,W: real,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W),M)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Z),W)))) ) ) ).

% norm_power_diff
tff(fact_4694_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z),M)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),M)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,Z),W)))) ) ) ).

% norm_power_diff
tff(fact_4695_lowi__def,axiom,
    ! [Xb: nat,N: nat] : aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Xb),N) = aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% lowi_def
tff(fact_4696_foldr__same,axiom,
    ! [Xs: list_real,Y: real] :
      ( ! [X2: real,Y2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => ( aa_set_real_o(member_real(Y2),set_real2(Xs))
           => ( X2 = Y2 ) ) )
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),set_real2(Xs))
           => ( X2 = Y ) )
       => ( aa_real_real(foldr_real_real(plus_plus_real,Xs),zero_zero_real) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,size_size_list_real(Xs))),Y) ) ) ) ).

% foldr_same
tff(fact_4697_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_VEBT_VEBT,Fa: fun_VEBT_VEBT_nat,Bound: nat,I: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(Fa,X2)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(Fa,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_4698_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Fa: fun_Pr7403558292872094996sn_nat,Bound: nat,I: nat] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro8055486230010607173sn_nat(Fa,X2)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_Pr7570552894071451325sn_nat(Fa,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6829681357464350627n_assn(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_4699_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_real,Fa: fun_real_nat,Bound: nat,I: nat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(Fa,X2)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_real_nat(Fa,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_4700_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_o,Fa: fun_o_nat,Bound: nat,I: nat] :
      ( ! [X2: $o] :
          ( member_o((X2),set_o2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(Fa,(X2))),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_o_nat(Fa,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_4701_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_nat,Fa: fun_nat_nat,Bound: nat,I: nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Fa,X2)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_nat_nat(Fa,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_4702_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_int,Fa: fun_int_nat,Bound: nat,I: nat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),set_int2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Fa,X2)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_int_nat(Fa,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_4703_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int),B2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_4704_of__nat__code__if,axiom,
    ! [N: nat] :
      aa_nat_uint32(semiri2565882477558803405uint32,N) = $ite(N = zero_zero_nat,zero_zero_uint32,produc2417093276151063866uint32(aTP_Lamm_mh,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_4705_of__nat__code__if,axiom,
    ! [N: nat] :
      aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,N) = $ite(N = zero_zero_nat,zero_z3563351764282998399l_num1,produc6192303373133366212l_num1(aTP_Lamm_mi,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_4706_of__nat__code__if,axiom,
    ! [N: nat] :
      aa_nat_rat(semiri681578069525770553at_rat,N) = $ite(N = zero_zero_nat,zero_zero_rat,produc6207742614233964070at_rat(aTP_Lamm_mj,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_4707_of__nat__code__if,axiom,
    ! [N: nat] :
      aa_nat_real(semiri5074537144036343181t_real,N) = $ite(N = zero_zero_nat,zero_zero_real,produc1703576794950452218t_real(aTP_Lamm_mk,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_4708_of__nat__code__if,axiom,
    ! [N: nat] :
      aa_nat_int(semiri1314217659103216013at_int,N) = $ite(N = zero_zero_nat,zero_zero_int,produc6840382203811409530at_int(aTP_Lamm_ml,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_4709_of__nat__code__if,axiom,
    ! [N: nat] :
      aa_nat_nat(semiri1316708129612266289at_nat,N) = $ite(N = zero_zero_nat,zero_zero_nat,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_mm),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_4710_of__nat__code__if,axiom,
    ! [N: nat] :
      aa_nat_Code_integer(semiri4939895301339042750nteger,N) = $ite(N = zero_zero_nat,zero_z3403309356797280102nteger,produc1830744345554046123nteger(aTP_Lamm_mn,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_4711_foldr0,axiom,
    ! [Xs: list_real,C: real,D: real] : aa_real_real(foldr_real_real(plus_plus_real,Xs),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(foldr_real_real(plus_plus_real,Xs),D)),C) ).

% foldr0
tff(fact_4712_mod__mod__trivial,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_mod_trivial
tff(fact_4713_mod__mod__trivial,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% mod_mod_trivial
tff(fact_4714_mod__mod__trivial,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% mod_mod_trivial
tff(fact_4715_low__def,axiom,
    ! [Xb: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% low_def
tff(fact_4716_map__ident,axiom,
    ! [X3: list_nat] : map_nat_nat(aTP_Lamm_eb,X3) = X3 ).

% map_ident
tff(fact_4717_bits__mod__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] : modulo1504961113040953224l_num1(zero_z3563351764282998399l_num1,A2) = zero_z3563351764282998399l_num1 ).

% bits_mod_0
tff(fact_4718_bits__mod__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A2) = zero_zero_nat ).

% bits_mod_0
tff(fact_4719_bits__mod__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A2) = zero_zero_int ).

% bits_mod_0
tff(fact_4720_bits__mod__0,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(zero_z3403309356797280102nteger),A2) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_4721_bits__mod__0,axiom,
    ! [A2: uint32] : aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,zero_zero_uint32),A2) = zero_zero_uint32 ).

% bits_mod_0
tff(fact_4722_mod__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A2) = zero_zero_nat ).

% mod_0
tff(fact_4723_mod__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A2) = zero_zero_int ).

% mod_0
tff(fact_4724_mod__0,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(zero_z3403309356797280102nteger),A2) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_4725_mod__by__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),zero_zero_nat) = A2 ).

% mod_by_0
tff(fact_4726_mod__by__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),zero_zero_int) = A2 ).

% mod_by_0
tff(fact_4727_mod__by__0,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),zero_z3403309356797280102nteger) = A2 ).

% mod_by_0
tff(fact_4728_mod__self,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),A2) = zero_zero_nat ).

% mod_self
tff(fact_4729_mod__self,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),A2) = zero_zero_int ).

% mod_self
tff(fact_4730_mod__self,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),A2) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_4731_mod__add__self1,axiom,
    ! [B2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_add_self1
tff(fact_4732_mod__add__self1,axiom,
    ! [B2: int,A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% mod_add_self1
tff(fact_4733_mod__add__self1,axiom,
    ! [B2: code_integer,A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),A2)),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% mod_add_self1
tff(fact_4734_mod__add__self2,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_add_self2
tff(fact_4735_mod__add__self2,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% mod_add_self2
tff(fact_4736_mod__add__self2,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% mod_add_self2
tff(fact_4737_minus__mod__self2,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% minus_mod_self2
tff(fact_4738_minus__mod__self2,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% minus_mod_self2
tff(fact_4739_length__map,axiom,
    ! [Fa: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT] : size_size_list_real(map_VEBT_VEBT_real(Fa,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_4740_length__map,axiom,
    ! [Fa: fun_real_real,Xs: list_real] : size_size_list_real(map_real_real(Fa,Xs)) = size_size_list_real(Xs) ).

% length_map
tff(fact_4741_length__map,axiom,
    ! [Fa: fun_o_real,Xs: list_o] : size_size_list_real(map_o_real(Fa,Xs)) = size_size_list_o(Xs) ).

% length_map
tff(fact_4742_length__map,axiom,
    ! [Fa: fun_nat_real,Xs: list_nat] : size_size_list_real(map_nat_real(Fa,Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_4743_length__map,axiom,
    ! [Fa: fun_int_real,Xs: list_int] : size_size_list_real(map_int_real(Fa,Xs)) = size_size_list_int(Xs) ).

% length_map
tff(fact_4744_length__map,axiom,
    ! [Fa: fun_real_o,Xs: list_real] : size_size_list_o(map_real_o(Fa,Xs)) = size_size_list_real(Xs) ).

% length_map
tff(fact_4745_length__map,axiom,
    ! [Fa: fun_o_o,Xs: list_o] : size_size_list_o(map_o_o(Fa,Xs)) = size_size_list_o(Xs) ).

% length_map
tff(fact_4746_length__map,axiom,
    ! [Fa: fun_nat_o,Xs: list_nat] : size_size_list_o(map_nat_o(Fa,Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_4747_length__map,axiom,
    ! [Fa: fun_int_o,Xs: list_int] : size_size_list_o(map_int_o(Fa,Xs)) = size_size_list_int(Xs) ).

% length_map
tff(fact_4748_length__map,axiom,
    ! [Fa: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT] : size_size_list_nat(map_VEBT_VEBT_nat(Fa,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_4749_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = M ) ) ).

% mod_less
tff(fact_4750_nat__mod__eq_H,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N) = A2 ) ) ).

% nat_mod_eq'
tff(fact_4751_bits__mod__by__1,axiom,
    ! [A2: word_N3645301735248828278l_num1] : modulo1504961113040953224l_num1(A2,one_on7727431528512463931l_num1) = zero_z3563351764282998399l_num1 ).

% bits_mod_by_1
tff(fact_4752_bits__mod__by__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),one_one_nat) = zero_zero_nat ).

% bits_mod_by_1
tff(fact_4753_bits__mod__by__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_4754_bits__mod__by__1,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_4755_bits__mod__by__1,axiom,
    ! [A2: uint32] : aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),one_one_uint32) = zero_zero_uint32 ).

% bits_mod_by_1
tff(fact_4756_mod__by__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),one_one_nat) = zero_zero_nat ).

% mod_by_1
tff(fact_4757_mod__by__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_4758_mod__by__1,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_4759_mod__mult__self1__is__0,axiom,
    ! [B2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),B2) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_4760_mod__mult__self1__is__0,axiom,
    ! [B2: int,A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)),B2) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_4761_mod__mult__self1__is__0,axiom,
    ! [B2: code_integer,A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),A2)),B2) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_4762_mod__mult__self2__is__0,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),B2) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_4763_mod__mult__self2__is__0,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),B2) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_4764_mod__mult__self2__is__0,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),B2) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_4765_bits__mod__div__trivial,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] : divide1791077408188789448l_num1(modulo1504961113040953224l_num1(A2,B2),B2) = zero_z3563351764282998399l_num1 ).

% bits_mod_div_trivial
tff(fact_4766_bits__mod__div__trivial,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),B2) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_4767_bits__mod__div__trivial,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),B2) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_4768_bits__mod__div__trivial,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),B2) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_4769_bits__mod__div__trivial,axiom,
    ! [A2: uint32,B2: uint32] : aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),B2)),B2) = zero_zero_uint32 ).

% bits_mod_div_trivial
tff(fact_4770_mod__div__trivial,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),B2) = zero_zero_nat ).

% mod_div_trivial
tff(fact_4771_mod__div__trivial,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),B2) = zero_zero_int ).

% mod_div_trivial
tff(fact_4772_mod__div__trivial,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),B2) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_4773_mod__mult__self4,axiom,
    ! [B2: nat,C: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_mult_self4
tff(fact_4774_mod__mult__self4,axiom,
    ! [B2: int,C: int,A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)),A2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% mod_mult_self4
tff(fact_4775_mod__mult__self4,axiom,
    ! [B2: code_integer,C: code_integer,A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)),A2)),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% mod_mult_self4
tff(fact_4776_mod__mult__self3,axiom,
    ! [C: nat,B2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2)),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_mult_self3
tff(fact_4777_mod__mult__self3,axiom,
    ! [C: int,B2: int,A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)),A2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% mod_mult_self3
tff(fact_4778_mod__mult__self3,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)),A2)),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% mod_mult_self3
tff(fact_4779_mod__mult__self2,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_mult_self2
tff(fact_4780_mod__mult__self2,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% mod_mult_self2
tff(fact_4781_mod__mult__self2,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C))),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% mod_mult_self2
tff(fact_4782_mod__mult__self1,axiom,
    ! [A2: nat,C: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_mult_self1
tff(fact_4783_mod__mult__self1,axiom,
    ! [A2: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% mod_mult_self1
tff(fact_4784_mod__mult__self1,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2))),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% mod_mult_self1
tff(fact_4785_mod__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_4786_fst__divmod__nat,axiom,
    ! [M: nat,N: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) ).

% fst_divmod_nat
tff(fact_4787_snd__divmod__nat,axiom,
    ! [M: nat,N: nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) ).

% snd_divmod_nat
tff(fact_4788_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Fa: fun_VE3165209603783583043T_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(map_VE8901447254227204932T_VEBT(Fa,Xs),N) = aa_VEB5832953974749077948T_VEBT(Fa,nth_VEBT_VEBT(Xs,N)) ) ) ).

% nth_map
tff(fact_4789_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Fa: fun_VE9120655849212850072T_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_VEBT_VEBT(map_VE7998069337340375161T_VEBT(Fa,Xs),N) = aa_VEB4941942288398148609T_VEBT(Fa,nth_VEBT_VEBTi(Xs,N)) ) ) ).

% nth_map
tff(fact_4790_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Fa: fun_VEBT_VEBTi_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_nat(nth_nat(map_VEBT_VEBTi_nat(Fa,Xs)),N) = aa_VEBT_VEBTi_nat(Fa,nth_VEBT_VEBTi(Xs,N)) ) ) ).

% nth_map
tff(fact_4791_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Fa: fun_VE712148701811721852_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBTi(map_VE7029150624388687525_VEBTi(Fa,Xs),N) = aa_VEB3973023575446460973_VEBTi(Fa,nth_VEBT_VEBT(Xs,N)) ) ) ).

% nth_map
tff(fact_4792_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Fa: fun_VE8449866691444886503_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_VEBT_VEBTi(map_VE483055756984248624_VEBTi(Fa,Xs),N) = aa_VEB5586453056816421800_VEBTi(Fa,nth_VEBT_VEBTi(Xs,N)) ) ) ).

% nth_map
tff(fact_4793_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Fa: fun_VEBT_VEBT_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_int(nth_int(map_VEBT_VEBT_int(Fa,Xs)),N) = aa_VEBT_VEBT_int(Fa,nth_VEBT_VEBT(Xs,N)) ) ) ).

% nth_map
tff(fact_4794_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Fa: fun_VEBT_VEBTi_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_int(nth_int(map_VEBT_VEBTi_int(Fa,Xs)),N) = aa_VEBT_VEBTi_int(Fa,nth_VEBT_VEBTi(Xs,N)) ) ) ).

% nth_map
tff(fact_4795_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Fa: fun_VEBT_VEBT_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_nat(nth_nat(map_VEBT_VEBT_nat(Fa,Xs)),N) = aa_VEBT_VEBT_nat(Fa,nth_VEBT_VEBT(Xs,N)) ) ) ).

% nth_map
tff(fact_4796_nth__map,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Fa: fun_VEBT_VEBT_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_real(nth_real(map_VEBT_VEBT_real(Fa,Xs)),N) = aa_VEBT_VEBT_real(Fa,nth_VEBT_VEBT(Xs,N)) ) ) ).

% nth_map
tff(fact_4797_nth__map,axiom,
    ! [N: nat,Xs: list_real,Fa: fun_real_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_real(Xs))
     => ( nth_VEBT_VEBT(map_real_VEBT_VEBT(Fa,Xs),N) = aa_real_VEBT_VEBT(Fa,aa_nat_real(nth_real(Xs),N)) ) ) ).

% nth_map
tff(fact_4798_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self4
tff(fact_4799_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)),M))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self3
tff(fact_4800_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),K)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self2
tff(fact_4801_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% Suc_mod_mult_self1
tff(fact_4802_numeral__mod__numeral,axiom,
    ! [K: num,L: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(K)),numera6620942414471956472nteger(L)) = produc6174133586879617921nteger(unique3479559517661332726nteger(K,L)) ).

% numeral_mod_numeral
tff(fact_4803_numeral__mod__numeral,axiom,
    ! [K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(K,L)) ).

% numeral_mod_numeral
tff(fact_4804_numeral__mod__numeral,axiom,
    ! [K: num,L: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,L)) = aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(K,L)) ).

% numeral_mod_numeral
tff(fact_4805_bits__one__mod__two__eq__one,axiom,
    modulo1504961113040953224l_num1(one_on7727431528512463931l_num1,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = one_on7727431528512463931l_num1 ).

% bits_one_mod_two_eq_one
tff(fact_4806_bits__one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% bits_one_mod_two_eq_one
tff(fact_4807_bits__one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_4808_bits__one__mod__two__eq__one,axiom,
    aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_one_mod_two_eq_one
tff(fact_4809_bits__one__mod__two__eq__one,axiom,
    aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,one_one_uint32),numera9087168376688890119uint32(aa_num_num(bit0,one))) = one_one_uint32 ).

% bits_one_mod_two_eq_one
tff(fact_4810_one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_mod_two_eq_one
tff(fact_4811_one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_4812_one__mod__two__eq__one,axiom,
    aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_4813_mod2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_4814_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),N))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_4815_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))) != zero_z3563351764282998399l_num1 )
    <=> ( modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = one_on7727431528512463931l_num1 ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4816_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4817_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4818_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4819_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: uint32] :
      ( ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))) != zero_zero_uint32 )
    <=> ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))) = one_one_uint32 ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4820_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))) != one_on7727431528512463931l_num1 )
    <=> ( modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = zero_z3563351764282998399l_num1 ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4821_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4822_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4823_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4824_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: uint32] :
      ( ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))) != one_one_uint32 )
    <=> ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))) = zero_zero_uint32 ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4825_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_4826_add__self__mod__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_4827_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,N)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_4828_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_4829_one__mod__numeral,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(one_one_Code_integer),numera6620942414471956472nteger(N)) = produc6174133586879617921nteger(unique3479559517661332726nteger(one,N)) ).

% one_mod_numeral
tff(fact_4830_one__mod__numeral,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,N)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,N)) ).

% one_mod_numeral
tff(fact_4831_one__mod__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,N)) = aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,N)) ).

% one_mod_numeral
tff(fact_4832_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_4833_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K)) ).

% Suc_0_mod_numeral
tff(fact_4834_list_Omap__ident,axiom,
    ! [Tb: list_nat] : map_nat_nat(aTP_Lamm_eb,Tb) = Tb ).

% list.map_ident
tff(fact_4835_of__nat__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% of_nat_mod
tff(fact_4836_of__nat__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N)) ).

% of_nat_mod
tff(fact_4837_of__nat__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N)) ).

% of_nat_mod
tff(fact_4838_map__eq__imp__length__eq,axiom,
    ! [Fa: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_nat,Ys: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_nat(Fa,Xs) = map_VEBT_VEBT_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_4839_map__eq__imp__length__eq,axiom,
    ! [Fa: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_real,Ys: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_real(Fa,Xs) = map_VEBT_VEBT_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_4840_map__eq__imp__length__eq,axiom,
    ! [Fa: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_real_nat,Ys: list_real] :
      ( ( map_VEBT_VEBT_nat(Fa,Xs) = map_real_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_4841_map__eq__imp__length__eq,axiom,
    ! [Fa: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_real_real,Ys: list_real] :
      ( ( map_VEBT_VEBT_real(Fa,Xs) = map_real_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_4842_map__eq__imp__length__eq,axiom,
    ! [Fa: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_o_nat,Ys: list_o] :
      ( ( map_VEBT_VEBT_nat(Fa,Xs) = map_o_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_o(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_4843_map__eq__imp__length__eq,axiom,
    ! [Fa: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_o_real,Ys: list_o] :
      ( ( map_VEBT_VEBT_real(Fa,Xs) = map_o_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_o(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_4844_map__eq__imp__length__eq,axiom,
    ! [Fa: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_nat_nat,Ys: list_nat] :
      ( ( map_VEBT_VEBT_nat(Fa,Xs) = map_nat_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_4845_map__eq__imp__length__eq,axiom,
    ! [Fa: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_nat_real,Ys: list_nat] :
      ( ( map_VEBT_VEBT_real(Fa,Xs) = map_nat_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_4846_map__eq__imp__length__eq,axiom,
    ! [Fa: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_int_nat,Ys: list_int] :
      ( ( map_VEBT_VEBT_nat(Fa,Xs) = map_int_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_4847_map__eq__imp__length__eq,axiom,
    ! [Fa: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_int_real,Ys: list_int] :
      ( ( map_VEBT_VEBT_real(Fa,Xs) = map_int_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_4848_map__eq__nth__eq,axiom,
    ! [Fa: fun_VEBT_VEBT_nat,L: list_VEBT_VEBT,L3: list_VEBT_VEBT,I: nat] :
      ( ( map_VEBT_VEBT_nat(Fa,L) = map_VEBT_VEBT_nat(Fa,L3) )
     => ( aa_VEBT_VEBT_nat(Fa,nth_VEBT_VEBT(L,I)) = aa_VEBT_VEBT_nat(Fa,nth_VEBT_VEBT(L3,I)) ) ) ).

% map_eq_nth_eq
tff(fact_4849_map__eq__nth__eq,axiom,
    ! [Fa: fun_VEBT_VEBT_real,L: list_VEBT_VEBT,L3: list_VEBT_VEBT,I: nat] :
      ( ( map_VEBT_VEBT_real(Fa,L) = map_VEBT_VEBT_real(Fa,L3) )
     => ( aa_VEBT_VEBT_real(Fa,nth_VEBT_VEBT(L,I)) = aa_VEBT_VEBT_real(Fa,nth_VEBT_VEBT(L3,I)) ) ) ).

% map_eq_nth_eq
tff(fact_4850_map__eq__nth__eq,axiom,
    ! [Fa: fun_nat_nat,L: list_nat,L3: list_nat,I: nat] :
      ( ( map_nat_nat(Fa,L) = map_nat_nat(Fa,L3) )
     => ( aa_nat_nat(Fa,aa_nat_nat(nth_nat(L),I)) = aa_nat_nat(Fa,aa_nat_nat(nth_nat(L3),I)) ) ) ).

% map_eq_nth_eq
tff(fact_4851_map__eq__nth__eq,axiom,
    ! [Fa: fun_nat_o,L: list_nat,L3: list_nat,I: nat] :
      ( ( map_nat_o(Fa,L) = map_nat_o(Fa,L3) )
     => ( aa_nat_o(Fa,aa_nat_nat(nth_nat(L),I))
      <=> aa_nat_o(Fa,aa_nat_nat(nth_nat(L3),I)) ) ) ).

% map_eq_nth_eq
tff(fact_4852_map__eq__nth__eq,axiom,
    ! [Fa: fun_Pr3816910165196019820n_assn,L: list_P8527749157015355191n_assn,L3: list_P8527749157015355191n_assn,I: nat] :
      ( ( map_Pr8991440229025900053n_assn(Fa,L) = map_Pr8991440229025900053n_assn(Fa,L3) )
     => ( aa_Pro2979818503705714077n_assn(Fa,nth_Pr1769885009046257848n_assn(L,I)) = aa_Pro2979818503705714077n_assn(Fa,nth_Pr1769885009046257848n_assn(L3,I)) ) ) ).

% map_eq_nth_eq
tff(fact_4853_mod__add__eq,axiom,
    ! [A2: nat,C: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C) ).

% mod_add_eq
tff(fact_4854_mod__add__eq,axiom,
    ! [A2: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) ).

% mod_add_eq
tff(fact_4855_mod__add__eq,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C))),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C) ).

% mod_add_eq
tff(fact_4856_mod__add__cong,axiom,
    ! [A2: nat,C: nat,A4: nat,B2: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B3)),C) ) ) ) ).

% mod_add_cong
tff(fact_4857_mod__add__cong,axiom,
    ! [A2: int,C: int,A4: int,B2: int,B3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A4),B3)),C) ) ) ) ).

% mod_add_cong
tff(fact_4858_mod__add__cong,axiom,
    ! [A2: code_integer,C: code_integer,A4: code_integer,B2: code_integer,B3: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A4),C) )
     => ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B3),C) )
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A4),B3)),C) ) ) ) ).

% mod_add_cong
tff(fact_4859_mod__add__left__eq,axiom,
    ! [A2: nat,C: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C)),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C) ).

% mod_add_left_eq
tff(fact_4860_mod__add__left__eq,axiom,
    ! [A2: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C)),B2)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) ).

% mod_add_left_eq
tff(fact_4861_mod__add__left__eq,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C)),B2)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C) ).

% mod_add_left_eq
tff(fact_4862_mod__add__right__eq,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C) ).

% mod_add_right_eq
tff(fact_4863_mod__add__right__eq,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) ).

% mod_add_right_eq
tff(fact_4864_mod__add__right__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C))),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C) ).

% mod_add_right_eq
tff(fact_4865_mod__mult__eq,axiom,
    ! [A2: nat,C: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C) ).

% mod_mult_eq
tff(fact_4866_mod__mult__eq,axiom,
    ! [A2: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C) ).

% mod_mult_eq
tff(fact_4867_mod__mult__eq,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C))),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),C) ).

% mod_mult_eq
tff(fact_4868_mod__mult__cong,axiom,
    ! [A2: nat,C: nat,A4: nat,B2: nat,B3: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_4869_mod__mult__cong,axiom,
    ! [A2: int,C: int,A4: int,B2: int,B3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A4),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_4870_mod__mult__cong,axiom,
    ! [A2: code_integer,C: code_integer,A4: code_integer,B2: code_integer,B3: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A4),C) )
     => ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B3),C) )
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A4),B3)),C) ) ) ) ).

% mod_mult_cong
tff(fact_4871_mod__mult__mult2,axiom,
    ! [A2: nat,C: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),C) ).

% mod_mult_mult2
tff(fact_4872_mod__mult__mult2,axiom,
    ! [A2: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),C) ).

% mod_mult_mult2
tff(fact_4873_mod__mult__mult2,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),C) ).

% mod_mult_mult2
tff(fact_4874_mult__mod__right,axiom,
    ! [C: nat,A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2)) ).

% mult_mod_right
tff(fact_4875_mult__mod__right,axiom,
    ! [C: int,A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) ).

% mult_mod_right
tff(fact_4876_mult__mod__right,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)) ).

% mult_mod_right
tff(fact_4877_mod__mult__left__eq,axiom,
    ! [A2: nat,C: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C)),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C) ).

% mod_mult_left_eq
tff(fact_4878_mod__mult__left__eq,axiom,
    ! [A2: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C)),B2)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C) ).

% mod_mult_left_eq
tff(fact_4879_mod__mult__left__eq,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C)),B2)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),C) ).

% mod_mult_left_eq
tff(fact_4880_mod__mult__right__eq,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C) ).

% mod_mult_right_eq
tff(fact_4881_mod__mult__right__eq,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C) ).

% mod_mult_right_eq
tff(fact_4882_mod__mult__right__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C))),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),C) ).

% mod_mult_right_eq
tff(fact_4883_mod__diff__eq,axiom,
    ! [A2: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C) ).

% mod_diff_eq
tff(fact_4884_mod__diff__eq,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C))),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),C) ).

% mod_diff_eq
tff(fact_4885_mod__diff__cong,axiom,
    ! [A2: int,C: int,A4: int,B2: int,B3: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A4),B3)),C) ) ) ) ).

% mod_diff_cong
tff(fact_4886_mod__diff__cong,axiom,
    ! [A2: code_integer,C: code_integer,A4: code_integer,B2: code_integer,B3: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A4),C) )
     => ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B3),C) )
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A4),B3)),C) ) ) ) ).

% mod_diff_cong
tff(fact_4887_mod__diff__left__eq,axiom,
    ! [A2: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C)),B2)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C) ).

% mod_diff_left_eq
tff(fact_4888_mod__diff__left__eq,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C)),B2)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),C) ).

% mod_diff_left_eq
tff(fact_4889_mod__diff__right__eq,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C) ).

% mod_diff_right_eq
tff(fact_4890_mod__diff__right__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C))),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),C) ).

% mod_diff_right_eq
tff(fact_4891_power__mod,axiom,
    ! [A2: nat,B2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),N)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),B2) ).

% power_mod
tff(fact_4892_power__mod,axiom,
    ! [A2: int,B2: int,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),N)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(A2),N)),B2) ).

% power_mod
tff(fact_4893_power__mod,axiom,
    ! [A2: code_integer,B2: code_integer,N: nat] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),N)),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),B2) ).

% power_mod
tff(fact_4894_mod__Suc__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) ).

% mod_Suc_eq
tff(fact_4895_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),N) ).

% mod_Suc_Suc_eq
tff(fact_4896_nat__mod__eq,axiom,
    ! [B2: nat,N: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),N)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),N) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),N) = B2 ) ) ) ).

% nat_mod_eq
tff(fact_4897_mod__plus__right,axiom,
    ! [A2: nat,Xb: nat,M: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),Xb)),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),Xb)),M) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),M) ) ) ).

% mod_plus_right
tff(fact_4898_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),M) ).

% mod_less_eq_dividend
tff(fact_4899_divmod__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,M),N) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ).

% divmod_nat_def
tff(fact_4900_unique__quotient,axiom,
    ! [A2: int,B2: int,Q2: int,R: int,Q5: int,R5: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A2,B2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A2,B2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R5))
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
tff(fact_4901_unique__remainder,axiom,
    ! [A2: int,B2: int,Q2: int,R: int,Q5: int,R5: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A2,B2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A2,B2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R5))
       => ( R = R5 ) ) ) ).

% unique_remainder
tff(fact_4902_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),A2) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4903_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),A2) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4904_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),A2) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_4905_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),B2) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4906_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),B2) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4907_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),B2) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_4908_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_4909_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_4910_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M))),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N))),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(Q2)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(N)),numera6620942414471956472nteger(Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_4911_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_4912_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_4913_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(one)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(N)),numera6620942414471956472nteger(one)) ).

% cong_exp_iff_simps(4)
tff(fact_4914_mod__eq__self__iff__div__eq__0,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) = A2 )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4915_mod__eq__self__iff__div__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = A2 )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4916_mod__eq__self__iff__div__eq__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) = A2 )
    <=> ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_4917_mod__eqE,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C) )
     => ~ ! [D2: int] : B2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,C),D2)) ) ).

% mod_eqE
tff(fact_4918_mod__eqE,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C) )
     => ~ ! [D2: code_integer] : B2 != aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D2)) ) ).

% mod_eqE
tff(fact_4919_div__add1__eq,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C))),C)) ).

% div_add1_eq
tff(fact_4920_div__add1__eq,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C))),C)) ).

% div_add1_eq
tff(fact_4921_div__add1__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),C)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),C))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C))),C)) ).

% div_add1_eq
tff(fact_4922_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),N) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) = N,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))) ).

% mod_Suc
tff(fact_4923_mod__induct,axiom,
    ! [P: fun_nat_o,N: nat,P2: nat,M: nat] :
      ( aa_nat_o(P,N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),P2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),P2)
         => ( ! [N3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N3),P2)
               => ( aa_nat_o(P,N3)
                 => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N3)),P2)) ) )
           => aa_nat_o(P,M) ) ) ) ) ).

% mod_induct
tff(fact_4924_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N) ) ).

% mod_less_divisor
tff(fact_4925_nat__mod__lem,axiom,
    ! [N: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),N)
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),N) = B2 ) ) ) ).

% nat_mod_lem
tff(fact_4926_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,N))),N) ).

% mod_Suc_le_divisor
tff(fact_4927_word__rot__lem,axiom,
    ! [L: nat,K: nat,D: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),L)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),L)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D),N)),L) = N ) ) ) ).

% word_rot_lem
tff(fact_4928_nat__minus__mod,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),M))),M) = zero_zero_nat ).

% nat_minus_mod
tff(fact_4929_mod__if,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N),M,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ).

% mod_if
tff(fact_4930_mod__nat__sub,axiom,
    ! [Xb: nat,Z: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Y) ) ) ).

% mod_nat_sub
tff(fact_4931_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

% mod_geq
tff(fact_4932_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
     => ? [Q3: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_4933_nat__minus__mod__plus__right,axiom,
    ! [N: nat,Xb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),M))),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),M) ).

% nat_minus_mod_plus_right
tff(fact_4934_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) ) ) ).

% le_mod_geq
tff(fact_4935_nat__mod__eq__iff,axiom,
    ! [Xb: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_4936_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B2)
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) = A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4937_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) = A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4938_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_4939_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4940_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4941_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_4942_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_4943_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_4944_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N))),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = zero_z3403309356797280102nteger )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(N)),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_4945_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_4946_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_4947_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(N)),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_4948_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_4949_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_4950_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M))),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(one)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_4951_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_4952_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_4953_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(one)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N))),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_4954_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_4955_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_4956_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M))),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N))),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(Q2)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(N)),numera6620942414471956472nteger(Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_4957_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_4958_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_4959_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M))),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,N))),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_4960_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_4961_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_4962_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M))),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N))),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_4963_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: nat,A2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C) ).

% cancel_div_mod_rules(2)
tff(fact_4964_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: int,A2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C) ).

% cancel_div_mod_rules(2)
tff(fact_4965_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2))),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C) ).

% cancel_div_mod_rules(2)
tff(fact_4966_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C) ).

% cancel_div_mod_rules(1)
tff(fact_4967_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),C) ).

% cancel_div_mod_rules(1)
tff(fact_4968_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),B2)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2))),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),C) ).

% cancel_div_mod_rules(1)
tff(fact_4969_mod__div__decomp,axiom,
    ! [A2: nat,B2: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) ).

% mod_div_decomp
tff(fact_4970_mod__div__decomp,axiom,
    ! [A2: int,B2: int] : A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) ).

% mod_div_decomp
tff(fact_4971_mod__div__decomp,axiom,
    ! [A2: code_integer,B2: code_integer] : A2 = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),B2)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) ).

% mod_div_decomp
tff(fact_4972_mod__div__decomp,axiom,
    ! [A2: uint32,B2: uint32] : A2 = plus_plus_uint32(aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),B2)),B2),aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),B2)) ).

% mod_div_decomp
tff(fact_4973_div__mult__mod__eq,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) = A2 ).

% div_mult_mod_eq
tff(fact_4974_div__mult__mod__eq,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) = A2 ).

% div_mult_mod_eq
tff(fact_4975_div__mult__mod__eq,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),B2)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) = A2 ).

% div_mult_mod_eq
tff(fact_4976_div__mult__mod__eq,axiom,
    ! [A2: uint32,B2: uint32] : plus_plus_uint32(aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),B2)),B2),aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),B2)) = A2 ).

% div_mult_mod_eq
tff(fact_4977_mod__div__mult__eq,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)) = A2 ).

% mod_div_mult_eq
tff(fact_4978_mod__div__mult__eq,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2)) = A2 ).

% mod_div_mult_eq
tff(fact_4979_mod__div__mult__eq,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),B2)) = A2 ).

% mod_div_mult_eq
tff(fact_4980_mod__div__mult__eq,axiom,
    ! [A2: uint32,B2: uint32] : plus_plus_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),B2),aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),B2)),B2)) = A2 ).

% mod_div_mult_eq
tff(fact_4981_mod__mult__div__eq,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2))) = A2 ).

% mod_mult_div_eq
tff(fact_4982_mod__mult__div__eq,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))) = A2 ).

% mod_mult_div_eq
tff(fact_4983_mod__mult__div__eq,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2))) = A2 ).

% mod_mult_div_eq
tff(fact_4984_mod__mult__div__eq,axiom,
    ! [A2: uint32,B2: uint32] : plus_plus_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),B2),aa_uint32_uint32(times_times_uint32(B2),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),B2))) = A2 ).

% mod_mult_div_eq
tff(fact_4985_mult__div__mod__eq,axiom,
    ! [B2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) = A2 ).

% mult_div_mod_eq
tff(fact_4986_mult__div__mod__eq,axiom,
    ! [B2: int,A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) = A2 ).

% mult_div_mod_eq
tff(fact_4987_mult__div__mod__eq,axiom,
    ! [B2: code_integer,A2: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) = A2 ).

% mult_div_mod_eq
tff(fact_4988_mult__div__mod__eq,axiom,
    ! [B2: uint32,A2: uint32] : plus_plus_uint32(aa_uint32_uint32(times_times_uint32(B2),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),B2)),aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),B2)) = A2 ).

% mult_div_mod_eq
tff(fact_4989_div__mult1__eq,axiom,
    ! [A2: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C))),C)) ).

% div_mult1_eq
tff(fact_4990_div__mult1__eq,axiom,
    ! [A2: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C))),C)) ).

% div_mult1_eq
tff(fact_4991_div__mult1__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),C))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C))),C)) ).

% div_mult1_eq
tff(fact_4992_minus__div__mult__eq__mod,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% minus_div_mult_eq_mod
tff(fact_4993_minus__div__mult__eq__mod,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% minus_div_mult_eq_mod
tff(fact_4994_minus__div__mult__eq__mod,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),B2)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% minus_div_mult_eq_mod
tff(fact_4995_minus__div__mult__eq__mod,axiom,
    ! [A2: uint32,B2: uint32] : minus_minus_uint32(A2,aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),B2)),B2)) = aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),B2) ).

% minus_div_mult_eq_mod
tff(fact_4996_minus__mod__eq__div__mult,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2) ).

% minus_mod_eq_div_mult
tff(fact_4997_minus__mod__eq__div__mult,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2) ).

% minus_mod_eq_div_mult
tff(fact_4998_minus__mod__eq__div__mult,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),B2) ).

% minus_mod_eq_div_mult
tff(fact_4999_minus__mod__eq__div__mult,axiom,
    ! [A2: uint32,B2: uint32] : minus_minus_uint32(A2,aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),B2)) = aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),B2)),B2) ).

% minus_mod_eq_div_mult
tff(fact_5000_minus__mod__eq__mult__div,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ).

% minus_mod_eq_mult_div
tff(fact_5001_minus__mod__eq__mult__div,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ).

% minus_mod_eq_mult_div
tff(fact_5002_minus__mod__eq__mult__div,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)) ).

% minus_mod_eq_mult_div
tff(fact_5003_minus__mod__eq__mult__div,axiom,
    ! [A2: uint32,B2: uint32] : minus_minus_uint32(A2,aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),B2)) = aa_uint32_uint32(times_times_uint32(B2),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),B2)) ).

% minus_mod_eq_mult_div
tff(fact_5004_minus__mult__div__eq__mod,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% minus_mult_div_eq_mod
tff(fact_5005_minus__mult__div__eq__mod,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ).

% minus_mult_div_eq_mod
tff(fact_5006_minus__mult__div__eq__mod,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2))) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ).

% minus_mult_div_eq_mod
tff(fact_5007_minus__mult__div__eq__mod,axiom,
    ! [A2: uint32,B2: uint32] : minus_minus_uint32(A2,aa_uint32_uint32(times_times_uint32(B2),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),B2))) = aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),B2) ).

% minus_mult_div_eq_mod
tff(fact_5008_zmde,axiom,
    ! [B2: int,A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) ).

% zmde
tff(fact_5009_zmde,axiom,
    ! [B2: code_integer,A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) ).

% zmde
tff(fact_5010_zmde,axiom,
    ! [B2: uint32,A2: uint32] : aa_uint32_uint32(times_times_uint32(B2),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),B2)) = minus_minus_uint32(A2,aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),B2)) ).

% zmde
tff(fact_5011_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)),N) ) ).

% mod_le_divisor
tff(fact_5012_div__less__mono,axiom,
    ! [A3: nat,B5: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),N) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B5),N) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),N)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B5),N)) ) ) ) ) ).

% div_less_mono
tff(fact_5013_mod__nat__add,axiom,
    ! [Xb: nat,Z: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y)),Z) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y)),Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y)),Z)) ) ) ) ).

% mod_nat_add
tff(fact_5014_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,N: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),N) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
       => ? [Q3: nat] : Xb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_5015_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => ~ ! [S2: nat] : N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_5016_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => ~ ! [S2: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_5017_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ).

% mod_mult2_eq
tff(fact_5018_div__mod__decomp,axiom,
    ! [A3: nat,N: nat] : A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),N)),N)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),N)) ).

% div_mod_decomp
tff(fact_5019_modulo__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),N)) ).

% modulo_nat_def
tff(fact_5020_eucl__rel__int__by0,axiom,
    ! [K: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K,zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K)) ).

% eucl_rel_int_by0
tff(fact_5021_snd__divmod,axiom,
    ! [M: num,N: num] : produc6174133586879617921nteger(unique3479559517661332726nteger(M,N)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N)) ).

% snd_divmod
tff(fact_5022_snd__divmod,axiom,
    ! [M: num,N: num] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(M,N)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)) ).

% snd_divmod
tff(fact_5023_snd__divmod,axiom,
    ! [M: num,N: num] : aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(M,N)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) ).

% snd_divmod
tff(fact_5024_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = Q2 ) ) ).

% div_int_unique
tff(fact_5025_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_5026_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_5027_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N))),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_5028_map__upd__eq,axiom,
    ! [I: nat,L: list_P8527749157015355191n_assn,Fa: fun_Pr3816910165196019820n_assn,Xb: produc6575502325842934193n_assn] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6829681357464350627n_assn(L))
       => ( aa_Pro2979818503705714077n_assn(Fa,nth_Pr1769885009046257848n_assn(L,I)) = aa_Pro2979818503705714077n_assn(Fa,Xb) ) )
     => ( map_Pr8991440229025900053n_assn(Fa,list_u4534839942911652127n_assn(L,I,Xb)) = map_Pr8991440229025900053n_assn(Fa,L) ) ) ).

% map_upd_eq
tff(fact_5029_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Fa: fun_VEBT_VEBT_nat,Xb: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
       => ( aa_VEBT_VEBT_nat(Fa,nth_VEBT_VEBT(L,I)) = aa_VEBT_VEBT_nat(Fa,Xb) ) )
     => ( map_VEBT_VEBT_nat(Fa,list_u1324408373059187874T_VEBT(L,I,Xb)) = map_VEBT_VEBT_nat(Fa,L) ) ) ).

% map_upd_eq
tff(fact_5030_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Fa: fun_VEBT_VEBT_real,Xb: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
       => ( aa_VEBT_VEBT_real(Fa,nth_VEBT_VEBT(L,I)) = aa_VEBT_VEBT_real(Fa,Xb) ) )
     => ( map_VEBT_VEBT_real(Fa,list_u1324408373059187874T_VEBT(L,I,Xb)) = map_VEBT_VEBT_real(Fa,L) ) ) ).

% map_upd_eq
tff(fact_5031_map__upd__eq,axiom,
    ! [I: nat,L: list_nat,Fa: fun_nat_nat,Xb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
       => ( aa_nat_nat(Fa,aa_nat_nat(nth_nat(L),I)) = aa_nat_nat(Fa,Xb) ) )
     => ( map_nat_nat(Fa,list_update_nat(L,I,Xb)) = map_nat_nat(Fa,L) ) ) ).

% map_upd_eq
tff(fact_5032_map__upd__eq,axiom,
    ! [I: nat,L: list_nat,Fa: fun_nat_o,Xb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
       => ( aa_nat_o(Fa,aa_nat_nat(nth_nat(L),I))
        <=> aa_nat_o(Fa,Xb) ) )
     => ( map_nat_o(Fa,list_update_nat(L,I,Xb)) = map_nat_o(Fa,L) ) ) ).

% map_upd_eq
tff(fact_5033_mod__mult2__eq_H,axiom,
    ! [A2: int,M: nat,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,N)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_5034_mod__mult2__eq_H,axiom,
    ! [A2: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,N)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).

% mod_mult2_eq'
tff(fact_5035_mod__mult2__eq_H,axiom,
    ! [A2: code_integer,M: nat,N: nat] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_Code_integer(semiri4939895301339042750nteger,N)))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).

% mod_mult2_eq'
tff(fact_5036_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = divide_divide_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% field_char_0_class.of_nat_div
tff(fact_5037_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)))),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% field_char_0_class.of_nat_div
tff(fact_5038_divmod__def,axiom,
    ! [M: num,N: num] : unique3479559517661332726nteger(M,N) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N)),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N))) ).

% divmod_def
tff(fact_5039_divmod__def,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ).

% divmod_def
tff(fact_5040_divmod__def,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(M,N) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).

% divmod_def
tff(fact_5041_split__mod,axiom,
    ! [P: fun_nat_o,M: nat,N: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))
    <=> ( ( ( N = zero_zero_nat )
         => aa_nat_o(P,M) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),N)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),I4)),J3) )
               => aa_nat_o(P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_5042_mod__lemma,axiom,
    ! [C: nat,R: nat,B2: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Q2),C))),R)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) ) ) ).

% mod_lemma
tff(fact_5043_real__of__nat__div__aux,axiom,
    ! [Xb: nat,D: nat] : aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),D))),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_5044_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L) )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_5045_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),C))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5046_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5047_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5048_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5049_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5050_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M))),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(one)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5051_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,N)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5052_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,N)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5053_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(one)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,N))),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(numera6620942414471956472nteger(N)),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5054_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),N) ).

% Suc_mod_eq_add3_mod
tff(fact_5055_divmod_H__nat__def,axiom,
    ! [M: num,N: num] : unique5055182867167087721od_nat(M,N) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))) ).

% divmod'_nat_def
tff(fact_5056_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_5057_divmod__digit__0_I2_J,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))),B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_5058_divmod__digit__0_I2_J,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),B2)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_5059_divmod__digit__0_I2_J,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2))),B2)
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_5060_bits__stable__imp__add__self,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = A2 )
     => ( plus_p361126936061061375l_num1(A2,modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one)))) = zero_z3563351764282998399l_num1 ) ) ).

% bits_stable_imp_add_self
tff(fact_5061_bits__stable__imp__add__self,axiom,
    ! [A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A2 )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_5062_bits__stable__imp__add__self,axiom,
    ! [A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A2 )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_5063_bits__stable__imp__add__self,axiom,
    ! [A2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) = A2 )
     => ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_5064_bits__stable__imp__add__self,axiom,
    ! [A2: uint32] :
      ( ( aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))) = A2 )
     => ( plus_plus_uint32(A2,aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one)))) = zero_zero_uint32 ) ) ).

% bits_stable_imp_add_self
tff(fact_5065_div__exp__mod__exp__eq,axiom,
    ! [A2: word_N3645301735248828278l_num1,N: nat,M: nat] : modulo1504961113040953224l_num1(divide1791077408188789448l_num1(A2,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),M)) = divide1791077408188789448l_num1(modulo1504961113040953224l_num1(A2,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M))),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_5066_div__exp__mod__exp__eq,axiom,
    ! [A2: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_5067_div__exp__mod__exp__eq,axiom,
    ! [A2: int,N: nat,M: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_5068_div__exp__mod__exp__eq,axiom,
    ! [A2: code_integer,N: nat,M: nat] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_5069_div__exp__mod__exp__eq,axiom,
    ! [A2: uint32,N: nat,M: nat] : aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),N))),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),M)) = aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)))),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),N)) ).

% div_exp_mod_exp_eq
tff(fact_5070_power__mod__div,axiom,
    ! [Xb: nat,N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))) ).

% power_mod_div
tff(fact_5071_verit__le__mono__div,axiom,
    ! [A3: nat,B5: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),N)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B5),N) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B5),N)) ) ) ).

% verit_le_mono_div
tff(fact_5072_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList)),zero_zero_nat)) ).

% VEBT_internal.cnt'.simps(2)
tff(fact_5073_divmod__digit__0_I1_J,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))),B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5074_divmod__digit__0_I1_J,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),B2)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5075_divmod__digit__0_I1_J,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2))),B2)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5076_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: word_N3645301735248828278l_num1] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( modulo1504961113040953224l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A2),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),M)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(modulo1504961113040953224l_num1(A2,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5077_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5078_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5079_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5080_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A2: uint32] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,aa_uint32_uint32(times_times_uint32(A2),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),M))),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),N)) = aa_uint32_uint32(times_times_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5081_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Xb) = Y )
     => ( ( ? [A: $o,B: $o] : Xb = vEBT_Leaf((A),(B))
         => ( Y != one_one_nat ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList2),Summary) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.cnt'.elims
tff(fact_5082_mod__double__modulus,axiom,
    ! [M: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
       => ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(Xb),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(Xb),M) )
          | ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(Xb),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(Xb),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5083_mod__double__modulus,axiom,
    ! [M: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5084_mod__double__modulus,axiom,
    ! [M: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),M) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5085_divmod__digit__1_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2)))
         => ( aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2))),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5086_divmod__digit__1_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5087_divmod__digit__1_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5088_set__bit__Suc,axiom,
    ! [N: nat,A2: word_N3645301735248828278l_num1] : bit_se4894374433684937756l_num1(aa_nat_nat(suc,N),A2) = plus_p361126936061061375l_num1(modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),bit_se4894374433684937756l_num1(N,divide1791077408188789448l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5089_set__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] : aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(aa_nat_nat(suc,N)),A2) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5090_set__bit__Suc,axiom,
    ! [N: nat,A2: uint32] : bit_se6647067497041451410uint32(aa_nat_nat(suc,N),A2) = plus_plus_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),bit_se6647067497041451410uint32(N,aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5091_set__bit__Suc,axiom,
    ! [N: nat,A2: int] : aa_int_int(bit_se7879613467334960850it_int(aa_nat_nat(suc,N)),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5092_set__bit__Suc,axiom,
    ! [N: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,N)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5093_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
    <=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q2)),R) )
        & $ite(
            aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L),
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,R),L) ),
            $ite(
              aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int),
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),R)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int) ),
              Q2 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_5094_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_space2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList)),zero_zero_nat)) ).

% VEBT_internal.space'.simps(2)
tff(fact_5095_divmod__digit__1_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2)))
         => ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2)))),one_one_Code_integer) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_5096_divmod__digit__1_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_5097_divmod__digit__1_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_5098_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Xb) = Y )
     => ( ( ? [A: $o,B: $o] : Xb = vEBT_Leaf((A),(B))
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList2),Summary) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.space'.elims
tff(fact_5099_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_space,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary2))),size_s6755466524823107622T_VEBT(TreeList))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList)),zero_zero_nat)) ).

% VEBT_internal.space.simps(2)
tff(fact_5100_VEBT__internal_Ospace_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,Xb) = Y )
     => ( ( ? [A: $o,B: $o] : Xb = vEBT_Leaf((A),(B))
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList2),Summary) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.space.elims
tff(fact_5101_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A2,B2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_5102_divmod__nat__if,axiom,
    ! [M: nat,N: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,M),N) = $ite(
        ( ( N = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N) ),
        aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),M),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_mo),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(divmod_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N)) ) ).

% divmod_nat_if
tff(fact_5103_div__half__nat,axiom,
    ! [Y: nat,Xb: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)) = $let(
            q: nat,
            q:= aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)),
            $let(
              r: nat,
              r:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,q),Y)),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),r),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,q),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,r),Y)),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,q),r)) ) ) ) ) ).

% div_half_nat
tff(fact_5104_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_5105_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_VEBT_VEBT,Fa: fun_VEBT_VEBT_real,Bound: real,I: real] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(Fa,X2)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(Fa,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,size_s6755466524823107622T_VEBT(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5106_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Fa: fun_Pr4198752369477672304n_real,Bound: real,I: real] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Pro6225755720812867233n_real(Fa,X2)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_Pr5018725648611240729n_real(Fa,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,size_s6829681357464350627n_assn(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5107_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_real,Fa: fun_real_real,Bound: real,I: real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(Fa,X2)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_real_real(Fa,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,size_size_list_real(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5108_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_o,Fa: fun_o_real,Bound: real,I: real] :
      ( ! [X2: $o] :
          ( member_o((X2),set_o2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(Fa,(X2))),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_o_real(Fa,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,size_size_list_o(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5109_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_nat,Fa: fun_nat_real,Bound: real,I: real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Fa,X2)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(Fa,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,size_size_list_nat(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5110_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_int,Fa: fun_int_real,Bound: real,I: real] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),set_int2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(Fa,X2)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_int_real(Fa,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,size_size_list_int(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5111_real__nat__list,axiom,
    ! [Fa: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,C: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(Fa,Xs)),C)) = aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(aTP_Lamm_mp(Fa),Xs)),aa_nat_real(semiri5074537144036343181t_real,C)) ).

% real_nat_list
tff(fact_5112_real__nat__list,axiom,
    ! [Fa: fun_nat_nat,Xs: list_nat,C: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_nat_nat(Fa,Xs)),C)) = aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(aTP_Lamm_mq(Fa),Xs)),aa_nat_real(semiri5074537144036343181t_real,C)) ).

% real_nat_list
tff(fact_5113_f__g__map__foldr__bound,axiom,
    ! [Xs: list_VEBT_VEBT,Fa: fun_VEBT_VEBT_real,C: real,G: fun_VEBT_VEBT_real,D: real] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(Fa,X2)),aa_real_real(times_times_real(C),aa_VEBT_VEBT_real(G,X2))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(Fa,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(C),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(G,Xs)),zero_zero_real))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_5114_f__g__map__foldr__bound,axiom,
    ! [Xs: list_nat,Fa: fun_nat_real,C: real,G: fun_nat_real,D: real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Fa,X2)),aa_real_real(times_times_real(C),aa_nat_real(G,X2))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(Fa,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(C),aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(G,Xs)),zero_zero_real))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_5115_f__g__map__foldr__bound,axiom,
    ! [Xs: list_real,Fa: fun_real_real,C: real,G: fun_real_real,D: real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(Fa,X2)),aa_real_real(times_times_real(C),aa_real_real(G,X2))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_real_real(Fa,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(C),aa_real_real(foldr_real_real(plus_plus_real,map_real_real(G,Xs)),zero_zero_real))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_5116_f__g__map__foldr__bound,axiom,
    ! [Xs: list_int,Fa: fun_int_real,C: real,G: fun_int_real,D: real] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),set_int2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(Fa,X2)),aa_real_real(times_times_real(C),aa_int_real(G,X2))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_int_real(Fa,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(C),aa_real_real(foldr_real_real(plus_plus_real,map_int_real(G,Xs)),zero_zero_real))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_5117_f__g__map__foldr__bound,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Fa: fun_Pr4198752369477672304n_real,C: real,G: fun_Pr4198752369477672304n_real,D: real] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Pro6225755720812867233n_real(Fa,X2)),aa_real_real(times_times_real(C),aa_Pro6225755720812867233n_real(G,X2))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_Pr5018725648611240729n_real(Fa,Xs)),D)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(C),aa_real_real(foldr_real_real(plus_plus_real,map_Pr5018725648611240729n_real(G,Xs)),zero_zero_real))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_5118_round__unique,axiom,
    ! [Xb: real,Y: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(Xb),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Y))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))))
       => ( aa_real_int(archim8280529875227126926d_real,Xb) = Y ) ) ) ).

% round_unique
tff(fact_5119_round__unique,axiom,
    ! [Xb: rat,Y: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(Xb,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),plus_plus_rat(Xb,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))))
       => ( archim7778729529865785530nd_rat(Xb) = Y ) ) ) ).

% round_unique
tff(fact_5120_mult__le__cancel__iff1,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_5121_mult__le__cancel__iff1,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_5122_mult__le__cancel__iff1,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_5123_mod__word__self,axiom,
    ! [W: word_N3645301735248828278l_num1] : modulo1504961113040953224l_num1(W,W) = zero_z3563351764282998399l_num1 ).

% mod_word_self
tff(fact_5124_listsum__bound,axiom,
    ! [Xs: list_VEBT_VEBT,Fa: fun_VEBT_VEBT_real,Y: real] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(Fa,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(Fa,Xs)),Y)) ) ).

% listsum_bound
tff(fact_5125_listsum__bound,axiom,
    ! [Xs: list_nat,Fa: fun_nat_real,Y: real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Fa,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(Fa,Xs)),Y)) ) ).

% listsum_bound
tff(fact_5126_listsum__bound,axiom,
    ! [Xs: list_real,Fa: fun_real_real,Y: real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Fa,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_real_real(Fa,Xs)),Y)) ) ).

% listsum_bound
tff(fact_5127_listsum__bound,axiom,
    ! [Xs: list_int,Fa: fun_int_real,Y: real] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),set_int2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Fa,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_int_real(Fa,Xs)),Y)) ) ).

% listsum_bound
tff(fact_5128_listsum__bound,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Fa: fun_Pr4198752369477672304n_real,Y: real] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Pro6225755720812867233n_real(Fa,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(foldr_real_real(plus_plus_real,map_Pr5018725648611240729n_real(Fa,Xs)),Y)) ) ).

% listsum_bound
tff(fact_5129_map__fst__mk__snd,axiom,
    ! [K: num,L: list_num] : map_Pr454908937103039467um_num(product_fst_num_num,map_nu2851882102140640437um_num(aTP_Lamm_mr(K),L)) = L ).

% map_fst_mk_snd
tff(fact_5130_map__fst__mk__snd,axiom,
    ! [K: produc4813437837504472865T_VEBT,L: list_nat] : map_Pr3018521781701129308BT_nat(produc758997459209783180T_VEBT,map_na3322839687800594908T_VEBT(aTP_Lamm_ms(K),L)) = L ).

% map_fst_mk_snd
tff(fact_5131_map__fst__mk__snd,axiom,
    ! [K: num,L: list_nat] : map_Pr5956769322976601943um_nat(product_fst_nat_num,map_na8006665559001981237at_num(aTP_Lamm_mt(K),L)) = L ).

% map_fst_mk_snd
tff(fact_5132_map__fst__mk__snd,axiom,
    ! [K: nat,L: list_nat] : map_Pr3938374229010428429at_nat(product_fst_nat_nat,map_na7298421622053143531at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_mu,K),L)) = L ).

% map_fst_mk_snd
tff(fact_5133_map__fst__mk__snd,axiom,
    ! [K: int,L: list_int] : map_Pr6494458893431244577nt_int(product_fst_int_int,map_in7157766398909135175nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_mv,K),L)) = L ).

% map_fst_mk_snd
tff(fact_5134_map__fst__mk__snd,axiom,
    ! [K: assn,L: list_assn] : map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,map_as2373307505041272643n_assn(aTP_Lamm_mw(K),L)) = L ).

% map_fst_mk_snd
tff(fact_5135_map__snd__mk__fst,axiom,
    ! [K: num,L: list_num] : map_Pr454908937103039467um_num(product_snd_num_num,map_nu2851882102140640437um_num(product_Pair_num_num(K),L)) = L ).

% map_snd_mk_fst
tff(fact_5136_map__snd__mk__fst,axiom,
    ! [K: nat,L: list_P6730324909620535719T_VEBT] : map_Pr3651661567648760725T_VEBT(produc2084898568784432842T_VEBT,map_Pr7909041933469137955T_VEBT(produc1750349459881913976T_VEBT(K),L)) = L ).

% map_snd_mk_fst
tff(fact_5137_map__snd__mk__fst,axiom,
    ! [K: nat,L: list_num] : map_Pr2514101109132380577um_num(product_snd_nat_num,map_nu4721551698833171051at_num(product_Pair_nat_num(K),L)) = L ).

% map_snd_mk_fst
tff(fact_5138_map__snd__mk__fst,axiom,
    ! [K: nat,L: list_nat] : map_Pr3938374229010428429at_nat(product_snd_nat_nat,map_na7298421622053143531at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,K),L)) = L ).

% map_snd_mk_fst
tff(fact_5139_map__snd__mk__fst,axiom,
    ! [K: int,L: list_int] : map_Pr6494458893431244577nt_int(product_snd_int_int,map_in7157766398909135175nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K),L)) = L ).

% map_snd_mk_fst
tff(fact_5140_map__snd__mk__fst,axiom,
    ! [K: assn,L: list_assn] : map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,map_as2373307505041272643n_assn(produc118845697133431529n_assn(K),L)) = L ).

% map_snd_mk_fst
tff(fact_5141_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_5142_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_5143_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_5144_round__0,axiom,
    archim7778729529865785530nd_rat(zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_5145_round__numeral,axiom,
    ! [N: num] : aa_real_int(archim8280529875227126926d_real,numeral_numeral_real(N)) = aa_num_int(numeral_numeral_int,N) ).

% round_numeral
tff(fact_5146_round__numeral,axiom,
    ! [N: num] : archim7778729529865785530nd_rat(numeral_numeral_rat(N)) = aa_num_int(numeral_numeral_int,N) ).

% round_numeral
tff(fact_5147_round__1,axiom,
    aa_real_int(archim8280529875227126926d_real,one_one_real) = one_one_int ).

% round_1
tff(fact_5148_round__1,axiom,
    archim7778729529865785530nd_rat(one_one_rat) = one_one_int ).

% round_1
tff(fact_5149_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W))) ).

% zmod_numeral_Bit0
tff(fact_5150_one__mod__exp__eq__one,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) = one_one_int ).

% one_mod_exp_eq_one
tff(fact_5151_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_5152_word__mod__by__0,axiom,
    ! [K: word_N3645301735248828278l_num1] : modulo1504961113040953224l_num1(K,zero_z3563351764282998399l_num1) = K ).

% word_mod_by_0
tff(fact_5153_zmod__helper,axiom,
    ! [N: int,M: int,K: int,A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),M) = K )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,N),A2)),M) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A2)),M) ) ) ).

% zmod_helper
tff(fact_5154_pair__list__eqI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( map_Pr3938374229010428429at_nat(product_fst_nat_nat,Xs) = map_Pr3938374229010428429at_nat(product_fst_nat_nat,Ys) )
     => ( ( map_Pr3938374229010428429at_nat(product_snd_nat_nat,Xs) = map_Pr3938374229010428429at_nat(product_snd_nat_nat,Ys) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
tff(fact_5155_pair__list__eqI,axiom,
    ! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( map_Pr6494458893431244577nt_int(product_fst_int_int,Xs) = map_Pr6494458893431244577nt_int(product_fst_int_int,Ys) )
     => ( ( map_Pr6494458893431244577nt_int(product_snd_int_int,Xs) = map_Pr6494458893431244577nt_int(product_snd_int_int,Ys) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
tff(fact_5156_pair__list__eqI,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,Xs) = map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,Ys) )
     => ( ( map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,Xs) = map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,Ys) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
tff(fact_5157_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),K)),M) ) ).

% zmod_le_nonneg_dividend
tff(fact_5158_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_5159_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) ) ).

% neg_mod_bound
tff(fact_5160_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),D) = zero_zero_int )
     => ? [Q3: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q3) ) ).

% zmod_eq_0D
tff(fact_5161_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),D) = zero_zero_int )
    <=> ? [Q4: int] : M = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q4) ) ).

% zmod_eq_0_iff
tff(fact_5162_word__mod__less__divisor,axiom,
    ! [N: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(zero_z3563351764282998399l_num1),N)
     => aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(modulo1504961113040953224l_num1(M,N)),N) ) ).

% word_mod_less_divisor
tff(fact_5163_zmod__int,axiom,
    ! [A2: nat,B2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% zmod_int
tff(fact_5164_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = R ) ) ).

% mod_int_unique
tff(fact_5165_round__mono,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim7778729529865785530nd_rat(Xb)),archim7778729529865785530nd_rat(Y)) ) ).

% round_mono
tff(fact_5166_int__mod__ge,axiom,
    ! [A2: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),N)) ) ) ).

% int_mod_ge
tff(fact_5167_int__mod__lem,axiom,
    ! [N: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),N)
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),N) )
      <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),N) = B2 ) ) ) ).

% int_mod_lem
tff(fact_5168_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I),K) = I )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_5169_int__mod__eq,axiom,
    ! [B2: int,N: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),N)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),N) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),N) )
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),N) = B2 ) ) ) ) ).

% int_mod_eq
tff(fact_5170_pos__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),B2) ) ) ).

% pos_mod_conj
tff(fact_5171_neg__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) ) ) ).

% neg_mod_conj
tff(fact_5172_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_5173_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_5174_int__mod__le_H,axiom,
    ! [B2: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),N))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),N)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),N)) ) ).

% int_mod_le'
tff(fact_5175_nonneg__mod__div,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ) ).

% nonneg_mod_div
tff(fact_5176_zdiv__mono__strict,axiom,
    ! [A3: int,B5: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),N)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),N) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B5),N) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B5),N)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_5177_div__mod__decomp__int,axiom,
    ! [A3: int,N: int] : A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),N)),N)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),N)) ).

% div_mod_decomp_int
tff(fact_5178_divmod__int__def,axiom,
    ! [M: num,N: num] : unique5052692396658037445od_int(M,N) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).

% divmod_int_def
tff(fact_5179_mod__div__equality__div__eq,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) ).

% mod_div_equality_div_eq
tff(fact_5180_ceiling__ge__round,axiom,
    ! [Xb: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim8280529875227126926d_real,Xb)),aa_real_int(archim7802044766580827645g_real,Xb)) ).

% ceiling_ge_round
tff(fact_5181_eucl__rel__int,axiom,
    ! [K: int,L: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ).

% eucl_rel_int
tff(fact_5182_pos__mod__bound2,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% pos_mod_bound2
tff(fact_5183_int__mod__ge_H,axiom,
    ! [B2: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),N)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),N)) ) ) ).

% int_mod_ge'
tff(fact_5184_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_5185_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_5186_real__of__int__div__aux,axiom,
    ! [Xb: int,D: int] : aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Xb)),aa_int_real(ring_1_of_int_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),D))),aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),D))),aa_int_real(ring_1_of_int_real,D))) ).

% real_of_int_div_aux
tff(fact_5187_pos__mod__sign2,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% pos_mod_sign2
tff(fact_5188_nmod2,axiom,
    ! [N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int )
      | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% nmod2
tff(fact_5189_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% mod_2_neq_1_eq_eq_0
tff(fact_5190_mod__exp__less__eq__exp,axiom,
    ! [A2: int,N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% mod_exp_less_eq_exp
tff(fact_5191_mod__power__lem,axiom,
    ! [A2: int,N: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(A2),M)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N),zero_zero_int,aa_nat_int(power_power_int(A2),N)) ) ) ).

% mod_power_lem
tff(fact_5192_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q2)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B2)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = R ) ) ) ) ).

% int_mod_pos_eq
tff(fact_5193_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q2)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),R)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = R ) ) ) ) ).

% int_mod_neg_eq
tff(fact_5194_split__zmod,axiom,
    ! [P: fun_int_o,N: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,N) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => aa_int_o(P,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I4: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
             => aa_int_o(P,J3) ) ) ) ) ).

% split_zmod
tff(fact_5195_mod__sub__if__z,axiom,
    ! [Xb: int,Z: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
             => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),Z) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),Z)) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_5196_mod__add__if__z,axiom,
    ! [Xb: int,Z: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
             => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),Z) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),Z)) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_5197_zmod__zmult2__eq,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_5198_axxmod2,axiom,
    ! [Xb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Xb)),Xb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int )
      & ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),Xb)),Xb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% axxmod2
tff(fact_5199_z1pmod2,axiom,
    ! [B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% z1pmod2
tff(fact_5200_verit__le__mono__div__int,axiom,
    ! [A3: int,B5: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),N)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B5),N) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B5),N)) ) ) ).

% verit_le_mono_div_int
tff(fact_5201_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K))
      <=> ! [I4: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
              & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I4),J3) ) ) ) ).

% split_pos_lemma
tff(fact_5202_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),K))
      <=> ! [I4: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
              & ( N = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I4)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I4),J3) ) ) ) ).

% split_neg_lemma
tff(fact_5203_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList),Summary2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary2))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList)),zero_zero_real)) ).

% VEBT_internal.cnt.simps(2)
tff(fact_5204_p1mod22k,axiom,
    ! [B2: int,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),one_one_int)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))),one_one_int) ).

% p1mod22k
tff(fact_5205_p1mod22k_H,axiom,
    ! [B2: int,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))) ).

% p1mod22k'
tff(fact_5206_VEBT__internal_Ocnt_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: real] :
      ( ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Xb) = Y )
     => ( ( ? [A: $o,B: $o] : Xb = vEBT_Leaf((A),(B))
         => ( Y != one_one_real ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList2),Summary) )
             => ( Y != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList2)),zero_zero_real)) ) ) ) ) ).

% VEBT_internal.cnt.elims
tff(fact_5207_pos__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),A2))) ) ) ).

% pos_zmod_mult_2
tff(fact_5208_sb__inc__lem,axiom,
    ! [A2: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))) ) ).

% sb_inc_lem
tff(fact_5209_neg__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),one_one_int)),A2))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_5210_mult__less__iff1,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_5211_mult__less__iff1,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_5212_mult__less__iff1,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_5213_of__int__round__le,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_5214_of__int__round__le,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xb))),plus_plus_rat(Xb,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_5215_of__int__round__ge,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Xb),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).

% of_int_round_ge
tff(fact_5216_of__int__round__ge,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(Xb,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xb))) ).

% of_int_round_ge
tff(fact_5217_of__int__round__gt,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(Xb),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).

% of_int_round_gt
tff(fact_5218_of__int__round__gt,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(Xb,divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xb))) ).

% of_int_round_gt
tff(fact_5219_mult__le__cancel__iff2,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Z),Xb)),aa_real_real(times_times_real(Z),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_5220_mult__le__cancel__iff2,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Xb)),aa_rat_rat(times_times_rat(Z),Y))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_5221_mult__le__cancel__iff2,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_5222_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
    ! [A_of_integer: fun_Co4416855287806144957at_nat,Zero: nat,I: code_integer] :
      aa_Code_integer_nat(code_T3174267988120873389le_nat(A_of_integer,Zero),I) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),zero_z3403309356797280102nteger),
        undefined_nat,
        $ite(
          I = zero_z3403309356797280102nteger,
          Zero,
          $ite(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(I),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_Cod330945565099897510at_nat(A_of_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(I),numera6620942414471956472nteger(aa_num_num(bit0,one))))),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_Cod330945565099897510at_nat(A_of_integer,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(I),numera6620942414471956472nteger(aa_num_num(bit0,one)))),one_one_Code_integer)))) ) ) ).

% quickcheck_narrowing_samples.partial_term_of_sample_def
tff(fact_5223_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
    ! [A_of_integer: fun_Co4113693717567439989nt_int,Zero: int,I: code_integer] :
      aa_Code_integer_int(code_T3171777517611823113le_int(A_of_integer,Zero),I) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),zero_z3403309356797280102nteger),
        undefined_int,
        $ite(
          I = zero_z3403309356797280102nteger,
          Zero,
          $ite(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(I),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_Cod6375820294346531166nt_int(A_of_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(I),numera6620942414471956472nteger(aa_num_num(bit0,one))))),aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_Cod6375820294346531166nt_int(A_of_integer,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(I),numera6620942414471956472nteger(aa_num_num(bit0,one)))),one_one_Code_integer)))) ) ) ).

% quickcheck_narrowing_samples.partial_term_of_sample_def
tff(fact_5224_divides__aux__eq,axiom,
    ! [Q2: nat,R: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Q2),R))
    <=> ( R = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_5225_divides__aux__eq,axiom,
    ! [Q2: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
    <=> ( R = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_5226_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [Xb: nat,Y: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != one_one_int ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != one_one_int ) )
         => ~ ! [N3: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
tff(fact_5227_refines__assert_H__bind,axiom,
    ! [P2: heap_T2636463487746394924on_nat,Q2: heap_T2636463487746394924on_nat,Phi: $o] :
      ( refine7594492741263601813on_nat(P2,Q2)
     => refine7594492741263601813on_nat(P2,heap_T5999496708990702694on_nat(refine_Imp_assert((Phi)),aTP_Lamm_km(Q2))) ) ).

% refines_assert'_bind
tff(fact_5228_refines__assert_H__bind,axiom,
    ! [P2: heap_Time_Heap_o,Q2: heap_Time_Heap_o,Phi: $o] :
      ( aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(P2),Q2)
     => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(P2),heap_T7577940988442287570unit_o(refine_Imp_assert((Phi)),aTP_Lamm_ko(Q2))) ) ).

% refines_assert'_bind
tff(fact_5229_refines__assert_H__bind,axiom,
    ! [P2: heap_T8145700208782473153_VEBTi,Q2: heap_T8145700208782473153_VEBTi,Phi: $o] :
      ( refine5565527176597971370_VEBTi(P2,Q2)
     => refine5565527176597971370_VEBTi(P2,heap_T7982501707604696571_VEBTi(refine_Imp_assert((Phi)),aTP_Lamm_kp(Q2))) ) ).

% refines_assert'_bind
tff(fact_5230_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_5231_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_5232_dvd__0__right,axiom,
    ! [A2: uint32] : aa_uint32_o(dvd_dvd_uint32(A2),zero_zero_uint32) ).

% dvd_0_right
tff(fact_5233_dvd__0__right,axiom,
    ! [A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),zero_z3403309356797280102nteger) ).

% dvd_0_right
tff(fact_5234_dvd__0__right,axiom,
    ! [A2: word_N3645301735248828278l_num1] : aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),zero_z3563351764282998399l_num1) ).

% dvd_0_right
tff(fact_5235_dvd__0__right,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),zero_zero_real) ).

% dvd_0_right
tff(fact_5236_dvd__0__right,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),zero_zero_rat) ).

% dvd_0_right
tff(fact_5237_dvd__0__right,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),zero_zero_nat) ).

% dvd_0_right
tff(fact_5238_dvd__0__right,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),zero_zero_int) ).

% dvd_0_right
tff(fact_5239_dvd__0__left__iff,axiom,
    ! [A2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(zero_zero_uint32),A2)
    <=> ( A2 = zero_zero_uint32 ) ) ).

% dvd_0_left_iff
tff(fact_5240_dvd__0__left__iff,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A2)
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
tff(fact_5241_dvd__0__left__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(zero_z3563351764282998399l_num1),A2)
    <=> ( A2 = zero_z3563351764282998399l_num1 ) ) ).

% dvd_0_left_iff
tff(fact_5242_dvd__0__left__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A2)
    <=> ( A2 = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_5243_dvd__0__left__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A2)
    <=> ( A2 = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_5244_dvd__0__left__iff,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A2)
    <=> ( A2 = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_5245_dvd__0__left__iff,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A2)
    <=> ( A2 = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_5246_dvd__add__triv__left__iff,axiom,
    ! [A2: uint32,B2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(A2),plus_plus_uint32(A2,B2))
    <=> aa_uint32_o(dvd_dvd_uint32(A2),B2) ) ).

% dvd_add_triv_left_iff
tff(fact_5247_dvd__add__triv__left__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),plus_p361126936061061375l_num1(A2,B2))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),B2) ) ).

% dvd_add_triv_left_iff
tff(fact_5248_dvd__add__triv__left__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2) ) ).

% dvd_add_triv_left_iff
tff(fact_5249_dvd__add__triv__left__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2) ) ).

% dvd_add_triv_left_iff
tff(fact_5250_dvd__add__triv__left__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),plus_plus_rat(A2,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2) ) ).

% dvd_add_triv_left_iff
tff(fact_5251_dvd__add__triv__left__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2) ) ).

% dvd_add_triv_left_iff
tff(fact_5252_dvd__add__triv__left__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ).

% dvd_add_triv_left_iff
tff(fact_5253_dvd__add__triv__right__iff,axiom,
    ! [A2: uint32,B2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(A2),plus_plus_uint32(B2,A2))
    <=> aa_uint32_o(dvd_dvd_uint32(A2),B2) ) ).

% dvd_add_triv_right_iff
tff(fact_5254_dvd__add__triv__right__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),plus_p361126936061061375l_num1(B2,A2))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),B2) ) ).

% dvd_add_triv_right_iff
tff(fact_5255_dvd__add__triv__right__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2) ) ).

% dvd_add_triv_right_iff
tff(fact_5256_dvd__add__triv__right__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2) ) ).

% dvd_add_triv_right_iff
tff(fact_5257_dvd__add__triv__right__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),plus_plus_rat(B2,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2) ) ).

% dvd_add_triv_right_iff
tff(fact_5258_dvd__add__triv__right__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2) ) ).

% dvd_add_triv_right_iff
tff(fact_5259_dvd__add__triv__right__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ).

% dvd_add_triv_right_iff
tff(fact_5260_div__dvd__div,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),C)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),A2)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),A2))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),C) ) ) ) ).

% div_dvd_div
tff(fact_5261_div__dvd__div,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A2))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C) ) ) ) ).

% div_dvd_div
tff(fact_5262_div__dvd__div,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A2))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C) ) ) ) ).

% div_dvd_div
tff(fact_5263_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),one_one_nat)
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_5264_dvd__times__right__cancel__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_5265_dvd__times__right__cancel__iff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_5266_dvd__times__right__cancel__iff,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_5267_dvd__times__left__cancel__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_5268_dvd__times__left__cancel__iff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_5269_dvd__times__left__cancel__iff,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A2),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_5270_dvd__mult__cancel__right,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2) ) ) ).

% dvd_mult_cancel_right
tff(fact_5271_dvd__mult__cancel__right,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),C))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2) ) ) ).

% dvd_mult_cancel_right
tff(fact_5272_dvd__mult__cancel__right,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2) ) ) ).

% dvd_mult_cancel_right
tff(fact_5273_dvd__mult__cancel__right,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ) ).

% dvd_mult_cancel_right
tff(fact_5274_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2) ) ) ).

% dvd_mult_cancel_left
tff(fact_5275_dvd__mult__cancel__left,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(C),A2)),aa_real_real(times_times_real(C),B2))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2) ) ) ).

% dvd_mult_cancel_left
tff(fact_5276_dvd__mult__cancel__left,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(C),A2)),aa_rat_rat(times_times_rat(C),B2))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2) ) ) ).

% dvd_mult_cancel_left
tff(fact_5277_dvd__mult__cancel__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ) ).

% dvd_mult_cancel_left
tff(fact_5278_dvd__add__times__triv__left__iff,axiom,
    ! [A2: uint32,C: uint32,B2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(A2),plus_plus_uint32(aa_uint32_uint32(times_times_uint32(C),A2),B2))
    <=> aa_uint32_o(dvd_dvd_uint32(A2),B2) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5279_dvd__add__times__triv__left__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),plus_p361126936061061375l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(C),A2),B2))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),B2) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5280_dvd__add__times__triv__left__iff,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)),B2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5281_dvd__add__times__triv__left__iff,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(C),A2)),B2))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5282_dvd__add__times__triv__left__iff,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),plus_plus_rat(aa_rat_rat(times_times_rat(C),A2),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5283_dvd__add__times__triv__left__iff,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5284_dvd__add__times__triv__left__iff,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)),B2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5285_dvd__add__times__triv__right__iff,axiom,
    ! [A2: uint32,B2: uint32,C: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(A2),plus_plus_uint32(B2,aa_uint32_uint32(times_times_uint32(C),A2)))
    <=> aa_uint32_o(dvd_dvd_uint32(A2),B2) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5286_dvd__add__times__triv__right__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),plus_p361126936061061375l_num1(B2,aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(C),A2)))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),B2) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5287_dvd__add__times__triv__right__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5288_dvd__add__times__triv__right__iff,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(times_times_real(C),A2)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5289_dvd__add__times__triv__right__iff,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),plus_plus_rat(B2,aa_rat_rat(times_times_rat(C),A2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5290_dvd__add__times__triv__right__iff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5291_dvd__add__times__triv__right__iff,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(times_times_int,C),A2)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5292_unit__prod,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),one_one_Code_integer) ) ) ).

% unit_prod
tff(fact_5293_unit__prod,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),one_one_nat) ) ) ).

% unit_prod
tff(fact_5294_unit__prod,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),one_one_int) ) ) ).

% unit_prod
tff(fact_5295_div__add,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),C)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),C)) ) ) ) ).

% div_add
tff(fact_5296_div__add,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)) ) ) ) ).

% div_add
tff(fact_5297_div__add,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)) ) ) ) ).

% div_add
tff(fact_5298_unit__div,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),one_one_Code_integer) ) ) ).

% unit_div
tff(fact_5299_unit__div,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),one_one_nat) ) ) ).

% unit_div
tff(fact_5300_unit__div,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),one_one_int) ) ) ).

% unit_div
tff(fact_5301_unit__div__1__unit,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),A2)),one_one_Code_integer) ) ).

% unit_div_1_unit
tff(fact_5302_unit__div__1__unit,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A2)),one_one_nat) ) ).

% unit_div_1_unit
tff(fact_5303_unit__div__1__unit,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A2)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_5304_unit__div__1__div__1,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),A2)) = A2 ) ) ).

% unit_div_1_div_1
tff(fact_5305_unit__div__1__div__1,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A2)) = A2 ) ) ).

% unit_div_1_div_1
tff(fact_5306_unit__div__1__div__1,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A2)) = A2 ) ) ).

% unit_div_1_div_1
tff(fact_5307_dvd__mult__div__cancel,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),A2)) = B2 ) ) ).

% dvd_mult_div_cancel
tff(fact_5308_dvd__mult__div__cancel,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2)) = B2 ) ) ).

% dvd_mult_div_cancel
tff(fact_5309_dvd__mult__div__cancel,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2)) = B2 ) ) ).

% dvd_mult_div_cancel
tff(fact_5310_dvd__div__mult__self,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),A2)),A2) = B2 ) ) ).

% dvd_div_mult_self
tff(fact_5311_dvd__div__mult__self,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2)),A2) = B2 ) ) ).

% dvd_div_mult_self
tff(fact_5312_dvd__div__mult__self,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2)),A2) = B2 ) ) ).

% dvd_div_mult_self
tff(fact_5313_div__diff,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)),C) = aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),C)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),C)) ) ) ) ).

% div_diff
tff(fact_5314_div__diff,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)) ) ) ) ).

% div_diff
tff(fact_5315_dvd__imp__mod__0,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),A2) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_5316_dvd__imp__mod__0,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),A2) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_5317_dvd__imp__mod__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2)
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),A2) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_5318_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(suc,zero_zero_nat))
    <=> ( M = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_5319_dvd__1__left,axiom,
    ! [K: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K) ).

% dvd_1_left
tff(fact_5320_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N))
    <=> ( ( K = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_5321_unit__mult__div__div,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),A2)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),A2) ) ) ).

% unit_mult_div_div
tff(fact_5322_unit__mult__div__div,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2) ) ) ).

% unit_mult_div_div
tff(fact_5323_unit__mult__div__div,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2) ) ) ).

% unit_mult_div_div
tff(fact_5324_unit__div__mult__self,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),A2)),A2) = B2 ) ) ).

% unit_div_mult_self
tff(fact_5325_unit__div__mult__self,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2)),A2) = B2 ) ) ).

% unit_div_mult_self
tff(fact_5326_unit__div__mult__self,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2)),A2) = B2 ) ) ).

% unit_div_mult_self
tff(fact_5327_odd__add,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ~ aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),plus_plus_uint32(A2,B2))
    <=> ~ ( ~ aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
        <=> ~ aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),B2) ) ) ).

% odd_add
tff(fact_5328_odd__add,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2))
    <=> ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2) ) ) ).

% odd_add
tff(fact_5329_odd__add,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),plus_p361126936061061375l_num1(A2,B2))
    <=> ~ ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
        <=> ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),B2) ) ) ).

% odd_add
tff(fact_5330_odd__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))
    <=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2) ) ) ).

% odd_add
tff(fact_5331_odd__add,axiom,
    ! [A2: int,B2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))
    <=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2) ) ) ).

% odd_add
tff(fact_5332_even__add,axiom,
    ! [A2: uint32,B2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),plus_plus_uint32(A2,B2))
    <=> ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
      <=> aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),B2) ) ) ).

% even_add
tff(fact_5333_even__add,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2) ) ) ).

% even_add
tff(fact_5334_even__add,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),plus_p361126936061061375l_num1(A2,B2))
    <=> ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
      <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),B2) ) ) ).

% even_add
tff(fact_5335_even__add,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2) ) ) ).

% even_add
tff(fact_5336_even__add,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2) ) ) ).

% even_add
tff(fact_5337_even__mult__iff,axiom,
    ! [A2: uint32,B2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(times_times_uint32(A2),B2))
    <=> ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
        | aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),B2) ) ) ).

% even_mult_iff
tff(fact_5338_even__mult__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2) ) ) ).

% even_mult_iff
tff(fact_5339_even__mult__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A2),B2))
    <=> ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
        | aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),B2) ) ) ).

% even_mult_iff
tff(fact_5340_even__mult__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2) ) ) ).

% even_mult_iff
tff(fact_5341_even__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2) ) ) ).

% even_mult_iff
tff(fact_5342_even__mod__2__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2) ) ).

% even_mod_2_iff
tff(fact_5343_even__mod__2__iff,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2) ) ).

% even_mod_2_iff
tff(fact_5344_even__mod__2__iff,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2) ) ).

% even_mod_2_iff
tff(fact_5345_even__mod__2__iff,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2) ) ).

% even_mod_2_iff
tff(fact_5346_even__mod__2__iff,axiom,
    ! [A2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))))
    <=> aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2) ) ).

% even_mod_2_iff
tff(fact_5347_even__Suc,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ) ).

% even_Suc
tff(fact_5348_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,N)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ) ).

% even_Suc_Suc_iff
tff(fact_5349_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(M)),numera6620942414471956472nteger(N))
    <=> unique5706413561485394159nteger(unique3479559517661332726nteger(N,M)) ) ).

% dvd_numeral_simp
tff(fact_5350_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))
    <=> aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,unique5055182867167087721od_nat(N,M)) ) ).

% dvd_numeral_simp
tff(fact_5351_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))
    <=> aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,unique5052692396658037445od_int(N,M)) ) ).

% dvd_numeral_simp
tff(fact_5352_even__plus__one__iff,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),one_one_Code_integer))
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2) ) ).

% even_plus_one_iff
tff(fact_5353_even__plus__one__iff,axiom,
    ! [A2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),plus_plus_uint32(A2,one_one_uint32))
    <=> ~ aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2) ) ).

% even_plus_one_iff
tff(fact_5354_even__plus__one__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),plus_p361126936061061375l_num1(A2,one_on7727431528512463931l_num1))
    <=> ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2) ) ).

% even_plus_one_iff
tff(fact_5355_even__plus__one__iff,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2) ) ).

% even_plus_one_iff
tff(fact_5356_even__plus__one__iff,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2) ) ).

% even_plus_one_iff
tff(fact_5357_even__diff,axiom,
    ! [A2: uint32,B2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),minus_minus_uint32(A2,B2))
    <=> aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),plus_plus_uint32(A2,B2)) ) ).

% even_diff
tff(fact_5358_even__diff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)) ) ).

% even_diff
tff(fact_5359_even__diff,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),minus_4019991460397169231l_num1(A2,B2))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),plus_p361126936061061375l_num1(A2,B2)) ) ).

% even_diff
tff(fact_5360_even__diff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)) ) ).

% even_diff
tff(fact_5361_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_5362_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_5363_odd__succ__div__two,axiom,
    ! [A2: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_5364_odd__succ__div__two,axiom,
    ! [A2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_5365_odd__succ__div__two,axiom,
    ! [A2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_5366_even__succ__div__two,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_5367_even__succ__div__two,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_5368_even__succ__div__two,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_5369_even__succ__div__2,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),A2)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_5370_even__succ__div__2,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
     => ( divide1791077408188789448l_num1(plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,A2),numera7442385471795722001l_num1(aa_num_num(bit0,one))) = divide1791077408188789448l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_5371_even__succ__div__2,axiom,
    ! [A2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
     => ( aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,plus_plus_uint32(one_one_uint32,A2)),numera9087168376688890119uint32(aa_num_num(bit0,one))) = aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_5372_even__succ__div__2,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_5373_even__succ__div__2,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A2)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_5374_even__power,axiom,
    ! [A2: uint32,N: nat] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_uint32(power_power_uint32(A2),N))
    <=> ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% even_power
tff(fact_5375_even__power,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% even_power
tff(fact_5376_even__power,axiom,
    ! [A2: word_N3645301735248828278l_num1,N: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A2),N))
    <=> ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% even_power
tff(fact_5377_even__power,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% even_power
tff(fact_5378_even__power,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(A2),N))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% even_power
tff(fact_5379_zero__le__power__eq__numeral,axiom,
    ! [A2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_5380_zero__le__power__eq__numeral,axiom,
    ! [A2: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_5381_zero__le__power__eq__numeral,axiom,
    ! [A2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_5382_zero__le__power__eq__numeral,axiom,
    ! [A2: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_5383_power__less__zero__eq,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),zero_z3403309356797280102nteger)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger) ) ) ).

% power_less_zero_eq
tff(fact_5384_power__less__zero__eq,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_5385_power__less__zero__eq,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),N)),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_5386_power__less__zero__eq,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),N)),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_5387_power__less__zero__eq__numeral,axiom,
    ! [A2: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,W))),zero_z3403309356797280102nteger)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger) ) ) ).

% power_less_zero_eq_numeral
tff(fact_5388_power__less__zero__eq__numeral,axiom,
    ! [A2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real) ) ) ).

% power_less_zero_eq_numeral
tff(fact_5389_power__less__zero__eq__numeral,axiom,
    ! [A2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_5390_power__less__zero__eq__numeral,axiom,
    ! [A2: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_5391_even__of__nat,axiom,
    ! [N: nat] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_uint32(semiri2565882477558803405uint32,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ) ).

% even_of_nat
tff(fact_5392_even__of__nat,axiom,
    ! [N: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ) ).

% even_of_nat
tff(fact_5393_even__of__nat,axiom,
    ! [N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ) ).

% even_of_nat
tff(fact_5394_even__of__nat,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ) ).

% even_of_nat
tff(fact_5395_even__of__nat,axiom,
    ! [N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ) ).

% even_of_nat
tff(fact_5396_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_5397_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ) ) ).

% even_diff_nat
tff(fact_5398_odd__two__times__div__two__succ,axiom,
    ! [A2: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
     => ( aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))))),one_one_Code_integer) = A2 ) ) ).

% odd_two_times_div_two_succ
tff(fact_5399_odd__two__times__div__two__succ,axiom,
    ! [A2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A2 ) ) ).

% odd_two_times_div_two_succ
tff(fact_5400_odd__two__times__div__two__succ,axiom,
    ! [A2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A2 ) ) ).

% odd_two_times_div_two_succ
tff(fact_5401_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),minus_minus_uint32(aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),N),one_one_uint32))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5402_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)),one_one_Code_integer))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5403_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),minus_4019991460397169231l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N),one_on7727431528512463931l_num1))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5404_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5405_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5406_zero__less__power__eq__numeral,axiom,
    ! [A2: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A2 != zero_z3403309356797280102nteger ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_5407_zero__less__power__eq__numeral,axiom,
    ! [A2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A2 != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_5408_zero__less__power__eq__numeral,axiom,
    ! [A2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A2 != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_5409_zero__less__power__eq__numeral,axiom,
    ! [A2: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A2 != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_5410_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_5411_power__le__zero__eq__numeral,axiom,
    ! [A2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A2 = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_5412_power__le__zero__eq__numeral,axiom,
    ! [A2: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,W))),zero_z3403309356797280102nteger)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A2 = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_5413_power__le__zero__eq__numeral,axiom,
    ! [A2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A2 = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_5414_power__le__zero__eq__numeral,axiom,
    ! [A2: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A2 = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_5415_even__succ__div__exp,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),A2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_5416_even__succ__div__exp,axiom,
    ! [A2: word_N3645301735248828278l_num1,N: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( divide1791077408188789448l_num1(plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,A2),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N)) = divide1791077408188789448l_num1(A2,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_5417_even__succ__div__exp,axiom,
    ! [A2: uint32,N: nat] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,plus_plus_uint32(one_one_uint32,A2)),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),N)) = aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_5418_even__succ__div__exp,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_5419_even__succ__div__exp,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_5420_even__succ__mod__exp,axiom,
    ! [A2: word_N3645301735248828278l_num1,N: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( modulo1504961113040953224l_num1(plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,A2),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N)) = plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,modulo1504961113040953224l_num1(A2,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5421_even__succ__mod__exp,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5422_even__succ__mod__exp,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5423_even__succ__mod__exp,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),A2)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5424_even__succ__mod__exp,axiom,
    ! [A2: uint32,N: nat] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,plus_plus_uint32(one_one_uint32,A2)),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),N)) = plus_plus_uint32(one_one_uint32,aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5425_dvd__refl,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),A2) ).

% dvd_refl
tff(fact_5426_dvd__refl,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),A2) ).

% dvd_refl
tff(fact_5427_dvd__refl,axiom,
    ! [A2: uint32] : aa_uint32_o(dvd_dvd_uint32(A2),A2) ).

% dvd_refl
tff(fact_5428_dvd__refl,axiom,
    ! [A2: word_N3645301735248828278l_num1] : aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),A2) ).

% dvd_refl
tff(fact_5429_dvd__refl,axiom,
    ! [A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),A2) ).

% dvd_refl
tff(fact_5430_dvd__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C) ) ) ).

% dvd_trans
tff(fact_5431_dvd__trans,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C) ) ) ).

% dvd_trans
tff(fact_5432_dvd__trans,axiom,
    ! [A2: uint32,B2: uint32,C: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(A2),B2)
     => ( aa_uint32_o(dvd_dvd_uint32(B2),C)
       => aa_uint32_o(dvd_dvd_uint32(A2),C) ) ) ).

% dvd_trans
tff(fact_5433_dvd__trans,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),B2)
     => ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(B2),C)
       => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),C) ) ) ).

% dvd_trans
tff(fact_5434_dvd__trans,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),C) ) ) ).

% dvd_trans
tff(fact_5435_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N),M)
       => ( M = N ) ) ) ).

% dvd_antisym
tff(fact_5436_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N) ) ).

% of_nat_dvd_iff
tff(fact_5437_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N) ) ).

% of_nat_dvd_iff
tff(fact_5438_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N) ) ).

% of_nat_dvd_iff
tff(fact_5439_dvd__field__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2)
    <=> ( ( A2 = zero_zero_real )
       => ( B2 = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_5440_dvd__field__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2)
    <=> ( ( A2 = zero_zero_rat )
       => ( B2 = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_5441_dvd__0__left,axiom,
    ! [A2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(zero_zero_uint32),A2)
     => ( A2 = zero_zero_uint32 ) ) ).

% dvd_0_left
tff(fact_5442_dvd__0__left,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),A2)
     => ( A2 = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
tff(fact_5443_dvd__0__left,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(zero_z3563351764282998399l_num1),A2)
     => ( A2 = zero_z3563351764282998399l_num1 ) ) ).

% dvd_0_left
tff(fact_5444_dvd__0__left,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A2)
     => ( A2 = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_5445_dvd__0__left,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A2)
     => ( A2 = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_5446_dvd__0__left,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A2)
     => ( A2 = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_5447_dvd__0__left,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A2)
     => ( A2 = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_5448_dvd__add,axiom,
    ! [A2: uint32,B2: uint32,C: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(A2),B2)
     => ( aa_uint32_o(dvd_dvd_uint32(A2),C)
       => aa_uint32_o(dvd_dvd_uint32(A2),plus_plus_uint32(B2,C)) ) ) ).

% dvd_add
tff(fact_5449_dvd__add,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),B2)
     => ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),C)
       => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),plus_p361126936061061375l_num1(B2,C)) ) ) ).

% dvd_add
tff(fact_5450_dvd__add,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),C)) ) ) ).

% dvd_add
tff(fact_5451_dvd__add,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),C)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C)) ) ) ).

% dvd_add
tff(fact_5452_dvd__add,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),C)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),plus_plus_rat(B2,C)) ) ) ).

% dvd_add
tff(fact_5453_dvd__add,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)) ) ) ).

% dvd_add
tff(fact_5454_dvd__add,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ) ) ).

% dvd_add
tff(fact_5455_dvd__add__left__iff,axiom,
    ! [A2: uint32,C: uint32,B2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(A2),C)
     => ( aa_uint32_o(dvd_dvd_uint32(A2),plus_plus_uint32(B2,C))
      <=> aa_uint32_o(dvd_dvd_uint32(A2),B2) ) ) ).

% dvd_add_left_iff
tff(fact_5456_dvd__add__left__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),C)
     => ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),plus_p361126936061061375l_num1(B2,C))
      <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),B2) ) ) ).

% dvd_add_left_iff
tff(fact_5457_dvd__add__left__iff,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2) ) ) ).

% dvd_add_left_iff
tff(fact_5458_dvd__add__left__iff,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),C)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2) ) ) ).

% dvd_add_left_iff
tff(fact_5459_dvd__add__left__iff,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),plus_plus_rat(B2,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2) ) ) ).

% dvd_add_left_iff
tff(fact_5460_dvd__add__left__iff,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2) ) ) ).

% dvd_add_left_iff
tff(fact_5461_dvd__add__left__iff,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ) ).

% dvd_add_left_iff
tff(fact_5462_dvd__add__right__iff,axiom,
    ! [A2: uint32,B2: uint32,C: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(A2),B2)
     => ( aa_uint32_o(dvd_dvd_uint32(A2),plus_plus_uint32(B2,C))
      <=> aa_uint32_o(dvd_dvd_uint32(A2),C) ) ) ).

% dvd_add_right_iff
tff(fact_5463_dvd__add__right__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),B2)
     => ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),plus_p361126936061061375l_num1(B2,C))
      <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),C) ) ) ).

% dvd_add_right_iff
tff(fact_5464_dvd__add__right__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(B2),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),C) ) ) ).

% dvd_add_right_iff
tff(fact_5465_dvd__add__right__iff,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),C) ) ) ).

% dvd_add_right_iff
tff(fact_5466_dvd__add__right__iff,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),plus_plus_rat(B2,C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),C) ) ) ).

% dvd_add_right_iff
tff(fact_5467_dvd__add__right__iff,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C) ) ) ).

% dvd_add_right_iff
tff(fact_5468_dvd__add__right__iff,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C) ) ) ).

% dvd_add_right_iff
tff(fact_5469_one__dvd,axiom,
    ! [A2: word_N3645301735248828278l_num1] : aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(one_on7727431528512463931l_num1),A2) ).

% one_dvd
tff(fact_5470_one__dvd,axiom,
    ! [A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,one_one_Code_integer),A2) ).

% one_dvd
tff(fact_5471_one__dvd,axiom,
    ! [A2: uint32] : aa_uint32_o(dvd_dvd_uint32(one_one_uint32),A2) ).

% one_dvd
tff(fact_5472_one__dvd,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,one_one_real),A2) ).

% one_dvd
tff(fact_5473_one__dvd,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,one_one_rat),A2) ).

% one_dvd
tff(fact_5474_one__dvd,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A2) ).

% one_dvd
tff(fact_5475_one__dvd,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A2) ).

% one_dvd
tff(fact_5476_unit__imp__dvd,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2) ) ).

% unit_imp_dvd
tff(fact_5477_unit__imp__dvd,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2) ) ).

% unit_imp_dvd
tff(fact_5478_unit__imp__dvd,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2) ) ).

% unit_imp_dvd
tff(fact_5479_dvd__unit__imp__unit,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer) ) ) ).

% dvd_unit_imp_unit
tff(fact_5480_dvd__unit__imp__unit,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat) ) ) ).

% dvd_unit_imp_unit
tff(fact_5481_dvd__unit__imp__unit,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int) ) ) ).

% dvd_unit_imp_unit
tff(fact_5482_dvdE,axiom,
    ! [B2: uint32,A2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(B2),A2)
     => ~ ! [K2: uint32] : A2 != aa_uint32_uint32(times_times_uint32(B2),K2) ) ).

% dvdE
tff(fact_5483_dvdE,axiom,
    ! [B2: word_N3645301735248828278l_num1,A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(B2),A2)
     => ~ ! [K2: word_N3645301735248828278l_num1] : A2 != aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B2),K2) ) ).

% dvdE
tff(fact_5484_dvdE,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2)
     => ~ ! [K2: code_integer] : A2 != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),K2) ) ).

% dvdE
tff(fact_5485_dvdE,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A2)
     => ~ ! [K2: real] : A2 != aa_real_real(times_times_real(B2),K2) ) ).

% dvdE
tff(fact_5486_dvdE,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A2)
     => ~ ! [K2: rat] : A2 != aa_rat_rat(times_times_rat(B2),K2) ) ).

% dvdE
tff(fact_5487_dvdE,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
     => ~ ! [K2: nat] : A2 != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),K2) ) ).

% dvdE
tff(fact_5488_dvdE,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
     => ~ ! [K2: int] : A2 != aa_int_int(aa_int_fun_int_int(times_times_int,B2),K2) ) ).

% dvdE
tff(fact_5489_dvdI,axiom,
    ! [A2: uint32,B2: uint32,K: uint32] :
      ( ( A2 = aa_uint32_uint32(times_times_uint32(B2),K) )
     => aa_uint32_o(dvd_dvd_uint32(B2),A2) ) ).

% dvdI
tff(fact_5490_dvdI,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( ( A2 = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B2),K) )
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(B2),A2) ) ).

% dvdI
tff(fact_5491_dvdI,axiom,
    ! [A2: code_integer,B2: code_integer,K: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2) ) ).

% dvdI
tff(fact_5492_dvdI,axiom,
    ! [A2: real,B2: real,K: real] :
      ( ( A2 = aa_real_real(times_times_real(B2),K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A2) ) ).

% dvdI
tff(fact_5493_dvdI,axiom,
    ! [A2: rat,B2: rat,K: rat] :
      ( ( A2 = aa_rat_rat(times_times_rat(B2),K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A2) ) ).

% dvdI
tff(fact_5494_dvdI,axiom,
    ! [A2: nat,B2: nat,K: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),K) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2) ) ).

% dvdI
tff(fact_5495_dvdI,axiom,
    ! [A2: int,B2: int,K: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(times_times_int,B2),K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2) ) ).

% dvdI
tff(fact_5496_dvd__def,axiom,
    ! [B2: uint32,A2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(B2),A2)
    <=> ? [K3: uint32] : A2 = aa_uint32_uint32(times_times_uint32(B2),K3) ) ).

% dvd_def
tff(fact_5497_dvd__def,axiom,
    ! [B2: word_N3645301735248828278l_num1,A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(B2),A2)
    <=> ? [K3: word_N3645301735248828278l_num1] : A2 = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B2),K3) ) ).

% dvd_def
tff(fact_5498_dvd__def,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2)
    <=> ? [K3: code_integer] : A2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),K3) ) ).

% dvd_def
tff(fact_5499_dvd__def,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A2)
    <=> ? [K3: real] : A2 = aa_real_real(times_times_real(B2),K3) ) ).

% dvd_def
tff(fact_5500_dvd__def,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A2)
    <=> ? [K3: rat] : A2 = aa_rat_rat(times_times_rat(B2),K3) ) ).

% dvd_def
tff(fact_5501_dvd__def,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
    <=> ? [K3: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),K3) ) ).

% dvd_def
tff(fact_5502_dvd__def,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
    <=> ? [K3: int] : A2 = aa_int_int(aa_int_fun_int_int(times_times_int,B2),K3) ) ).

% dvd_def
tff(fact_5503_dvd__mult,axiom,
    ! [A2: uint32,C: uint32,B2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(A2),C)
     => aa_uint32_o(dvd_dvd_uint32(A2),aa_uint32_uint32(times_times_uint32(B2),C)) ) ).

% dvd_mult
tff(fact_5504_dvd__mult,axiom,
    ! [A2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),C)
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B2),C)) ) ).

% dvd_mult
tff(fact_5505_dvd__mult,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) ) ).

% dvd_mult
tff(fact_5506_dvd__mult,axiom,
    ! [A2: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(times_times_real(B2),C)) ) ).

% dvd_mult
tff(fact_5507_dvd__mult,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(times_times_rat(B2),C)) ) ).

% dvd_mult
tff(fact_5508_dvd__mult,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) ) ).

% dvd_mult
tff(fact_5509_dvd__mult,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ) ).

% dvd_mult
tff(fact_5510_dvd__mult2,axiom,
    ! [A2: uint32,B2: uint32,C: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(A2),B2)
     => aa_uint32_o(dvd_dvd_uint32(A2),aa_uint32_uint32(times_times_uint32(B2),C)) ) ).

% dvd_mult2
tff(fact_5511_dvd__mult2,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),B2)
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B2),C)) ) ).

% dvd_mult2
tff(fact_5512_dvd__mult2,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) ) ).

% dvd_mult2
tff(fact_5513_dvd__mult2,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(times_times_real(B2),C)) ) ).

% dvd_mult2
tff(fact_5514_dvd__mult2,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(times_times_rat(B2),C)) ) ).

% dvd_mult2
tff(fact_5515_dvd__mult2,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) ) ).

% dvd_mult2
tff(fact_5516_dvd__mult2,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ) ).

% dvd_mult2
tff(fact_5517_dvd__mult__left,axiom,
    ! [A2: uint32,B2: uint32,C: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(aa_uint32_uint32(times_times_uint32(A2),B2)),C)
     => aa_uint32_o(dvd_dvd_uint32(A2),C) ) ).

% dvd_mult_left
tff(fact_5518_dvd__mult__left,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A2),B2)),C)
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),C) ) ).

% dvd_mult_left
tff(fact_5519_dvd__mult__left,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),C) ) ).

% dvd_mult_left
tff(fact_5520_dvd__mult__left,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(A2),B2)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),C) ) ).

% dvd_mult_left
tff(fact_5521_dvd__mult__left,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A2),B2)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),C) ) ).

% dvd_mult_left
tff(fact_5522_dvd__mult__left,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C) ) ).

% dvd_mult_left
tff(fact_5523_dvd__mult__left,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C) ) ).

% dvd_mult_left
tff(fact_5524_dvd__triv__left,axiom,
    ! [A2: uint32,B2: uint32] : aa_uint32_o(dvd_dvd_uint32(A2),aa_uint32_uint32(times_times_uint32(A2),B2)) ).

% dvd_triv_left
tff(fact_5525_dvd__triv__left,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] : aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A2),B2)) ).

% dvd_triv_left
tff(fact_5526_dvd__triv__left,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)) ).

% dvd_triv_left
tff(fact_5527_dvd__triv__left,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(times_times_real(A2),B2)) ).

% dvd_triv_left
tff(fact_5528_dvd__triv__left,axiom,
    ! [A2: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(times_times_rat(A2),B2)) ).

% dvd_triv_left
tff(fact_5529_dvd__triv__left,axiom,
    ! [A2: nat,B2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)) ).

% dvd_triv_left
tff(fact_5530_dvd__triv__left,axiom,
    ! [A2: int,B2: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) ).

% dvd_triv_left
tff(fact_5531_mult__dvd__mono,axiom,
    ! [A2: uint32,B2: uint32,C: uint32,D: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(A2),B2)
     => ( aa_uint32_o(dvd_dvd_uint32(C),D)
       => aa_uint32_o(dvd_dvd_uint32(aa_uint32_uint32(times_times_uint32(A2),C)),aa_uint32_uint32(times_times_uint32(B2),D)) ) ) ).

% mult_dvd_mono
tff(fact_5532_mult__dvd__mono,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,D: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),B2)
     => ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(C),D)
       => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A2),C)),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B2),D)) ) ) ).

% mult_dvd_mono
tff(fact_5533_mult__dvd__mono,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),D)) ) ) ).

% mult_dvd_mono
tff(fact_5534_mult__dvd__mono,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),D)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(A2),C)),aa_real_real(times_times_real(B2),D)) ) ) ).

% mult_dvd_mono
tff(fact_5535_mult__dvd__mono,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A2),C)),aa_rat_rat(times_times_rat(B2),D)) ) ) ).

% mult_dvd_mono
tff(fact_5536_mult__dvd__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D)) ) ) ).

% mult_dvd_mono
tff(fact_5537_mult__dvd__mono,axiom,
    ! [A2: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ).

% mult_dvd_mono
tff(fact_5538_dvd__mult__right,axiom,
    ! [A2: uint32,B2: uint32,C: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(aa_uint32_uint32(times_times_uint32(A2),B2)),C)
     => aa_uint32_o(dvd_dvd_uint32(B2),C) ) ).

% dvd_mult_right
tff(fact_5539_dvd__mult__right,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A2),B2)),C)
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(B2),C) ) ).

% dvd_mult_right
tff(fact_5540_dvd__mult__right,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),C) ) ).

% dvd_mult_right
tff(fact_5541_dvd__mult__right,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(A2),B2)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),C) ) ).

% dvd_mult_right
tff(fact_5542_dvd__mult__right,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A2),B2)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),C) ) ).

% dvd_mult_right
tff(fact_5543_dvd__mult__right,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C) ) ).

% dvd_mult_right
tff(fact_5544_dvd__mult__right,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C) ) ).

% dvd_mult_right
tff(fact_5545_dvd__triv__right,axiom,
    ! [A2: uint32,B2: uint32] : aa_uint32_o(dvd_dvd_uint32(A2),aa_uint32_uint32(times_times_uint32(B2),A2)) ).

% dvd_triv_right
tff(fact_5546_dvd__triv__right,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] : aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B2),A2)) ).

% dvd_triv_right
tff(fact_5547_dvd__triv__right,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),A2)) ).

% dvd_triv_right
tff(fact_5548_dvd__triv__right,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(times_times_real(B2),A2)) ).

% dvd_triv_right
tff(fact_5549_dvd__triv__right,axiom,
    ! [A2: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(times_times_rat(B2),A2)) ).

% dvd_triv_right
tff(fact_5550_dvd__triv__right,axiom,
    ! [A2: nat,B2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)) ).

% dvd_triv_right
tff(fact_5551_dvd__triv__right,axiom,
    ! [A2: int,B2: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)) ).

% dvd_triv_right
tff(fact_5552_dvd__diff,axiom,
    ! [Xb: uint32,Y: uint32,Z: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(Xb),Y)
     => ( aa_uint32_o(dvd_dvd_uint32(Xb),Z)
       => aa_uint32_o(dvd_dvd_uint32(Xb),minus_minus_uint32(Y,Z)) ) ) ).

% dvd_diff
tff(fact_5553_dvd__diff,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1,Z: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Xb),Y)
     => ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Xb),Z)
       => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Xb),minus_4019991460397169231l_num1(Y,Z)) ) ) ).

% dvd_diff
tff(fact_5554_dvd__diff,axiom,
    ! [Xb: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Z)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Y),Z)) ) ) ).

% dvd_diff
tff(fact_5555_dvd__diff,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Z)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_real_real(minus_minus_real(Y),Z)) ) ) ).

% dvd_diff
tff(fact_5556_dvd__diff,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Z)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),minus_minus_rat(Y,Z)) ) ) ).

% dvd_diff
tff(fact_5557_dvd__diff,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Z)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ) ) ).

% dvd_diff
tff(fact_5558_dvd__diff__commute,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(C),B2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B2),C)) ) ).

% dvd_diff_commute
tff(fact_5559_dvd__diff__commute,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) ) ).

% dvd_diff_commute
tff(fact_5560_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
       => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),C) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),C) )
        <=> ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_5561_dvd__div__eq__iff,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B2)
       => ( ( aa_real_real(divide_divide_real(A2),C) = aa_real_real(divide_divide_real(B2),C) )
        <=> ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_5562_dvd__div__eq__iff,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B2)
       => ( ( divide_divide_rat(A2,C) = divide_divide_rat(B2,C) )
        <=> ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_5563_dvd__div__eq__iff,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C) )
        <=> ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_5564_dvd__div__eq__iff,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C) )
        <=> ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_5565_dvd__div__eq__cancel,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),C) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),C) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_5566_dvd__div__eq__cancel,axiom,
    ! [A2: real,C: real,B2: real] :
      ( ( aa_real_real(divide_divide_real(A2),C) = aa_real_real(divide_divide_real(B2),C) )
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A2)
       => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B2)
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_5567_dvd__div__eq__cancel,axiom,
    ! [A2: rat,C: rat,B2: rat] :
      ( ( divide_divide_rat(A2,C) = divide_divide_rat(B2,C) )
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B2)
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_5568_dvd__div__eq__cancel,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_5569_dvd__div__eq__cancel,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C) )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A2)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_5570_div__div__div__same,axiom,
    ! [D: code_integer,B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),D)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),D)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2) ) ) ) ).

% div_div_div_same
tff(fact_5571_div__div__div__same,axiom,
    ! [D: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),D)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) ) ) ) ).

% div_div_div_same
tff(fact_5572_div__div__div__same,axiom,
    ! [D: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),D)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ) ) ) ).

% div_div_div_same
tff(fact_5573_dvd__power__same,axiom,
    ! [Xb: uint32,Y: uint32,N: nat] :
      ( aa_uint32_o(dvd_dvd_uint32(Xb),Y)
     => aa_uint32_o(dvd_dvd_uint32(aa_nat_uint32(power_power_uint32(Xb),N)),aa_nat_uint32(power_power_uint32(Y),N)) ) ).

% dvd_power_same
tff(fact_5574_dvd__power__same,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1,N: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Xb),Y)
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),N)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Y),N)) ) ).

% dvd_power_same
tff(fact_5575_dvd__power__same,axiom,
    ! [Xb: nat,Y: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N)) ) ).

% dvd_power_same
tff(fact_5576_dvd__power__same,axiom,
    ! [Xb: int,Y: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),N)),aa_nat_int(power_power_int(Y),N)) ) ).

% dvd_power_same
tff(fact_5577_dvd__power__same,axiom,
    ! [Xb: real,Y: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N)) ) ).

% dvd_power_same
tff(fact_5578_dvd__power__same,axiom,
    ! [Xb: complex,Y: complex,N: nat] :
      ( aa_complex_o(dvd_dvd_complex(Xb),Y)
     => aa_complex_o(dvd_dvd_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y),N)) ) ).

% dvd_power_same
tff(fact_5579_dvd__power__same,axiom,
    ! [Xb: code_integer,Y: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),N)) ) ).

% dvd_power_same
tff(fact_5580_dvd__mod__iff,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A2) ) ) ).

% dvd_mod_iff
tff(fact_5581_dvd__mod__iff,axiom,
    ! [C: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A2) ) ) ).

% dvd_mod_iff
tff(fact_5582_dvd__mod__iff,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A2) ) ) ).

% dvd_mod_iff
tff(fact_5583_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A2) ) ) ).

% dvd_mod_imp_dvd
tff(fact_5584_dvd__mod__imp__dvd,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A2) ) ) ).

% dvd_mod_imp_dvd
tff(fact_5585_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A2) ) ) ).

% dvd_mod_imp_dvd
tff(fact_5586_dvd__mod,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),N)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N)) ) ) ).

% dvd_mod
tff(fact_5587_dvd__mod,axiom,
    ! [K: int,M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),M)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),N)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),N)) ) ) ).

% dvd_mod
tff(fact_5588_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,N: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),N)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(M),N)) ) ) ).

% dvd_mod
tff(fact_5589_mod__mod__cancel,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C) ) ) ).

% mod_mod_cancel
tff(fact_5590_mod__mod__cancel,axiom,
    ! [C: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C) ) ) ).

% mod_mod_cancel
tff(fact_5591_mod__mod__cancel,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C) ) ) ).

% mod_mod_cancel
tff(fact_5592_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),N)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).

% dvd_diff_nat
tff(fact_5593_subset__divisors__dvd,axiom,
    ! [A2: complex,B2: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aTP_Lamm_mx(A2))),collect_complex(aTP_Lamm_mx(B2)))
    <=> aa_complex_o(dvd_dvd_complex(A2),B2) ) ).

% subset_divisors_dvd
tff(fact_5594_subset__divisors__dvd,axiom,
    ! [A2: int,B2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_my,A2))),collect_int(aa_int_fun_int_o(aTP_Lamm_my,B2)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ).

% subset_divisors_dvd
tff(fact_5595_subset__divisors__dvd,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ord_le2219237028632753026uint32(collect_uint32(aTP_Lamm_mz(A2)),collect_uint32(aTP_Lamm_mz(B2)))
    <=> aa_uint32_o(dvd_dvd_uint32(A2),B2) ) ).

% subset_divisors_dvd
tff(fact_5596_subset__divisors__dvd,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ord_le5203802739334966412l_num1(collec7814023847061821259l_num1(aTP_Lamm_na(A2)),collec7814023847061821259l_num1(aTP_Lamm_na(B2)))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),B2) ) ).

% subset_divisors_dvd
tff(fact_5597_subset__divisors__dvd,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_nb,A2))),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_nb,B2)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2) ) ).

% subset_divisors_dvd
tff(fact_5598_subset__divisors__dvd,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_nc,A2))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_nc,B2)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2) ) ).

% subset_divisors_dvd
tff(fact_5599_strict__subset__divisors__dvd,axiom,
    ! [A2: complex,B2: complex] :
      ( aa_set_complex_o(ord_less_set_complex(collect_complex(aTP_Lamm_mx(A2))),collect_complex(aTP_Lamm_mx(B2)))
    <=> ( aa_complex_o(dvd_dvd_complex(A2),B2)
        & ~ aa_complex_o(dvd_dvd_complex(B2),A2) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5600_strict__subset__divisors__dvd,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_set_nat_o(ord_less_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_nc,A2))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_nc,B2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5601_strict__subset__divisors__dvd,axiom,
    ! [A2: int,B2: int] :
      ( aa_set_int_o(ord_less_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_my,A2))),collect_int(aa_int_fun_int_o(aTP_Lamm_my,B2)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5602_strict__subset__divisors__dvd,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ord_less_set_uint32(collect_uint32(aTP_Lamm_mz(A2)),collect_uint32(aTP_Lamm_mz(B2)))
    <=> ( aa_uint32_o(dvd_dvd_uint32(A2),B2)
        & ~ aa_uint32_o(dvd_dvd_uint32(B2),A2) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5603_strict__subset__divisors__dvd,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ord_le6726900395242856064l_num1(collec7814023847061821259l_num1(aTP_Lamm_na(A2)),collec7814023847061821259l_num1(aTP_Lamm_na(B2)))
    <=> ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(A2),B2)
        & ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(B2),A2) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5604_strict__subset__divisors__dvd,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_nb,A2))),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_nb,B2)))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5605_not__is__unit__0,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% not_is_unit_0
tff(fact_5606_not__is__unit__0,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),one_one_nat) ).

% not_is_unit_0
tff(fact_5607_not__is__unit__0,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),one_one_int) ).

% not_is_unit_0
tff(fact_5608_minf_I10_J,axiom,
    ! [D: uint32,S: uint32] :
    ? [Z2: uint32] :
    ! [X3: uint32] :
      ( aa_uint32_o(ord_less_uint32(X3),Z2)
     => ( ~ aa_uint32_o(dvd_dvd_uint32(D),plus_plus_uint32(X3,S))
      <=> ~ aa_uint32_o(dvd_dvd_uint32(D),plus_plus_uint32(X3,S)) ) ) ).

% minf(10)
tff(fact_5609_minf_I10_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S: word_N3645301735248828278l_num1] :
    ? [Z2: word_N3645301735248828278l_num1] :
    ! [X3: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(X3),Z2)
     => ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,S))
      <=> ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,S)) ) ) ).

% minf(10)
tff(fact_5610_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Z2)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),S))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),S)) ) ) ).

% minf(10)
tff(fact_5611_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S)) ) ) ).

% minf(10)
tff(fact_5612_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X3,S))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X3,S)) ) ) ).

% minf(10)
tff(fact_5613_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)) ) ) ).

% minf(10)
tff(fact_5614_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S)) ) ) ).

% minf(10)
tff(fact_5615_minf_I9_J,axiom,
    ! [D: uint32,S: uint32] :
    ? [Z2: uint32] :
    ! [X3: uint32] :
      ( aa_uint32_o(ord_less_uint32(X3),Z2)
     => ( aa_uint32_o(dvd_dvd_uint32(D),plus_plus_uint32(X3,S))
      <=> aa_uint32_o(dvd_dvd_uint32(D),plus_plus_uint32(X3,S)) ) ) ).

% minf(9)
tff(fact_5616_minf_I9_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S: word_N3645301735248828278l_num1] :
    ? [Z2: word_N3645301735248828278l_num1] :
    ! [X3: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(X3),Z2)
     => ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,S))
      <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,S)) ) ) ).

% minf(9)
tff(fact_5617_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Z2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),S))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),S)) ) ) ).

% minf(9)
tff(fact_5618_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S)) ) ) ).

% minf(9)
tff(fact_5619_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X3,S))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X3,S)) ) ) ).

% minf(9)
tff(fact_5620_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)) ) ) ).

% minf(9)
tff(fact_5621_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S)) ) ) ).

% minf(9)
tff(fact_5622_pinf_I10_J,axiom,
    ! [D: uint32,S: uint32] :
    ? [Z2: uint32] :
    ! [X3: uint32] :
      ( aa_uint32_o(ord_less_uint32(Z2),X3)
     => ( ~ aa_uint32_o(dvd_dvd_uint32(D),plus_plus_uint32(X3,S))
      <=> ~ aa_uint32_o(dvd_dvd_uint32(D),plus_plus_uint32(X3,S)) ) ) ).

% pinf(10)
tff(fact_5623_pinf_I10_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S: word_N3645301735248828278l_num1] :
    ? [Z2: word_N3645301735248828278l_num1] :
    ! [X3: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Z2),X3)
     => ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,S))
      <=> ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,S)) ) ) ).

% pinf(10)
tff(fact_5624_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z2),X3)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),S))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),S)) ) ) ).

% pinf(10)
tff(fact_5625_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S)) ) ) ).

% pinf(10)
tff(fact_5626_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X3,S))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X3,S)) ) ) ).

% pinf(10)
tff(fact_5627_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)) ) ) ).

% pinf(10)
tff(fact_5628_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S)) ) ) ).

% pinf(10)
tff(fact_5629_pinf_I9_J,axiom,
    ! [D: uint32,S: uint32] :
    ? [Z2: uint32] :
    ! [X3: uint32] :
      ( aa_uint32_o(ord_less_uint32(Z2),X3)
     => ( aa_uint32_o(dvd_dvd_uint32(D),plus_plus_uint32(X3,S))
      <=> aa_uint32_o(dvd_dvd_uint32(D),plus_plus_uint32(X3,S)) ) ) ).

% pinf(9)
tff(fact_5630_pinf_I9_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S: word_N3645301735248828278l_num1] :
    ? [Z2: word_N3645301735248828278l_num1] :
    ! [X3: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Z2),X3)
     => ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,S))
      <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(D),plus_p361126936061061375l_num1(X3,S)) ) ) ).

% pinf(9)
tff(fact_5631_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z2),X3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),S))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X3),S)) ) ) ).

% pinf(9)
tff(fact_5632_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S)) ) ) ).

% pinf(9)
tff(fact_5633_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X3,S))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),plus_plus_rat(X3,S)) ) ) ).

% pinf(9)
tff(fact_5634_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S)) ) ) ).

% pinf(9)
tff(fact_5635_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S)) ) ) ).

% pinf(9)
tff(fact_5636_dvd__div__eq__0__iff,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2)
     => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2) = zero_z3403309356797280102nteger )
      <=> ( A2 = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5637_dvd__div__eq__0__iff,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A2)
     => ( ( aa_real_real(divide_divide_real(A2),B2) = zero_zero_real )
      <=> ( A2 = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5638_dvd__div__eq__0__iff,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A2)
     => ( ( divide_divide_rat(A2,B2) = zero_zero_rat )
      <=> ( A2 = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5639_dvd__div__eq__0__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) = zero_zero_nat )
      <=> ( A2 = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5640_dvd__div__eq__0__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = zero_zero_int )
      <=> ( A2 = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5641_unit__mult__right__cancel,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2) )
      <=> ( B2 = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_5642_unit__mult__right__cancel,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2) )
      <=> ( B2 = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_5643_unit__mult__right__cancel,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,C),A2) )
      <=> ( B2 = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_5644_unit__mult__left__cancel,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C) )
      <=> ( B2 = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_5645_unit__mult__left__cancel,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C) )
      <=> ( B2 = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_5646_unit__mult__left__cancel,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),C) )
      <=> ( B2 = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_5647_mult__unit__dvd__iff_H,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_5648_mult__unit__dvd__iff_H,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_5649_mult__unit__dvd__iff_H,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_5650_dvd__mult__unit__iff_H,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_5651_dvd__mult__unit__iff_H,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_5652_dvd__mult__unit__iff_H,axiom,
    ! [B2: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_5653_mult__unit__dvd__iff,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_5654_mult__unit__dvd__iff,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_5655_mult__unit__dvd__iff,axiom,
    ! [B2: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_5656_dvd__mult__unit__iff,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_5657_dvd__mult__unit__iff,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_5658_dvd__mult__unit__iff,axiom,
    ! [B2: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_5659_is__unit__mult__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer) ) ) ).

% is_unit_mult_iff
tff(fact_5660_is__unit__mult__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_5661_is__unit__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_5662_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A2)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),C)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_5663_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_5664_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_5665_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),C) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),C)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_5666_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_5667_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_5668_unit__div__cancel,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),A2) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),A2) )
      <=> ( B2 = C ) ) ) ).

% unit_div_cancel
tff(fact_5669_unit__div__cancel,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A2) )
      <=> ( B2 = C ) ) ) ).

% unit_div_cancel
tff(fact_5670_unit__div__cancel,axiom,
    ! [A2: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A2) )
      <=> ( B2 = C ) ) ) ).

% unit_div_cancel
tff(fact_5671_div__unit__dvd__iff,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),C) ) ) ).

% div_unit_dvd_iff
tff(fact_5672_div__unit__dvd__iff,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C) ) ) ).

% div_unit_dvd_iff
tff(fact_5673_div__unit__dvd__iff,axiom,
    ! [B2: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C) ) ) ).

% div_unit_dvd_iff
tff(fact_5674_dvd__div__unit__iff,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),B2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),C) ) ) ).

% dvd_div_unit_iff
tff(fact_5675_dvd__div__unit__iff,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C) ) ) ).

% dvd_div_unit_iff
tff(fact_5676_dvd__div__unit__iff,axiom,
    ! [B2: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C) ) ) ).

% dvd_div_unit_iff
tff(fact_5677_dvd__div__mult,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),C)),A2) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),A2)),C) ) ) ).

% dvd_div_mult
tff(fact_5678_dvd__div__mult,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),C) ) ) ).

% dvd_div_mult
tff(fact_5679_dvd__div__mult,axiom,
    ! [C: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)),A2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)),C) ) ) ).

% dvd_div_mult
tff(fact_5680_div__mult__swap,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),C) ) ) ).

% div_mult_swap
tff(fact_5681_div__mult__swap,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C) ) ) ).

% div_mult_swap
tff(fact_5682_div__mult__swap,axiom,
    ! [C: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C) ) ) ).

% div_mult_swap
tff(fact_5683_div__div__eq__right,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),C) ) ) ) ).

% div_div_eq_right
tff(fact_5684_div__div__eq__right,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C) ) ) ) ).

% div_div_eq_right
tff(fact_5685_div__div__eq__right,axiom,
    ! [C: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C) ) ) ) ).

% div_div_eq_right
tff(fact_5686_dvd__div__mult2__eq,axiom,
    ! [B2: code_integer,C: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)),A2)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_5687_dvd__div__mult2__eq,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_5688_dvd__div__mult2__eq,axiom,
    ! [B2: int,C: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)),A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_5689_dvd__mult__imp__div,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),C)) ) ).

% dvd_mult_imp_div
tff(fact_5690_dvd__mult__imp__div,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),B2)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)) ) ).

% dvd_mult_imp_div
tff(fact_5691_dvd__mult__imp__div,axiom,
    ! [A2: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),B2)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)) ) ).

% dvd_mult_imp_div
tff(fact_5692_div__mult__div__if__dvd,axiom,
    ! [B2: code_integer,A2: code_integer,D: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),C)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),D)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_5693_div__mult__div__if__dvd,axiom,
    ! [B2: nat,A2: nat,D: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),C)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_5694_div__mult__div__if__dvd,axiom,
    ! [B2: int,A2: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),C)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_5695_div__power,axiom,
    ! [B2: code_integer,A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),N) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N)) ) ) ).

% div_power
tff(fact_5696_div__power,axiom,
    ! [B2: nat,A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)) ) ) ).

% div_power
tff(fact_5697_div__power,axiom,
    ! [B2: int,A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
     => ( aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),N) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(B2),N)) ) ) ).

% div_power
tff(fact_5698_mod__0__imp__dvd,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( modulo1504961113040953224l_num1(A2,B2) = zero_z3563351764282998399l_num1 )
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(B2),A2) ) ).

% mod_0_imp_dvd
tff(fact_5699_mod__0__imp__dvd,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) = zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2) ) ).

% mod_0_imp_dvd
tff(fact_5700_mod__0__imp__dvd,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2) ) ).

% mod_0_imp_dvd
tff(fact_5701_mod__0__imp__dvd,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) = zero_z3403309356797280102nteger )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2) ) ).

% mod_0_imp_dvd
tff(fact_5702_mod__0__imp__dvd,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),B2) = zero_zero_uint32 )
     => aa_uint32_o(dvd_dvd_uint32(B2),A2) ) ).

% mod_0_imp_dvd
tff(fact_5703_dvd__eq__mod__eq__0,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),A2) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_5704_dvd__eq__mod__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),A2) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_5705_dvd__eq__mod__eq__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2)
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),A2) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_5706_mod__eq__0__iff__dvd,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2) ) ).

% mod_eq_0_iff_dvd
tff(fact_5707_mod__eq__0__iff__dvd,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2) ) ).

% mod_eq_0_iff_dvd
tff(fact_5708_mod__eq__0__iff__dvd,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2) ) ).

% mod_eq_0_iff_dvd
tff(fact_5709_dvd__power__le,axiom,
    ! [Xb: uint32,Y: uint32,N: nat,M: nat] :
      ( aa_uint32_o(dvd_dvd_uint32(Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => aa_uint32_o(dvd_dvd_uint32(aa_nat_uint32(power_power_uint32(Xb),N)),aa_nat_uint32(power_power_uint32(Y),M)) ) ) ).

% dvd_power_le
tff(fact_5710_dvd__power__le,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1,N: nat,M: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),N)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Y),M)) ) ) ).

% dvd_power_le
tff(fact_5711_dvd__power__le,axiom,
    ! [Xb: nat,Y: nat,N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),M)) ) ) ).

% dvd_power_le
tff(fact_5712_dvd__power__le,axiom,
    ! [Xb: int,Y: int,N: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),N)),aa_nat_int(power_power_int(Y),M)) ) ) ).

% dvd_power_le
tff(fact_5713_dvd__power__le,axiom,
    ! [Xb: real,Y: real,N: nat,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),M)) ) ) ).

% dvd_power_le
tff(fact_5714_dvd__power__le,axiom,
    ! [Xb: complex,Y: complex,N: nat,M: nat] :
      ( aa_complex_o(dvd_dvd_complex(Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => aa_complex_o(dvd_dvd_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y),M)) ) ) ).

% dvd_power_le
tff(fact_5715_dvd__power__le,axiom,
    ! [Xb: code_integer,Y: code_integer,N: nat,M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),N)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),M)) ) ) ).

% dvd_power_le
tff(fact_5716_power__le__dvd,axiom,
    ! [A2: uint32,N: nat,B2: uint32,M: nat] :
      ( aa_uint32_o(dvd_dvd_uint32(aa_nat_uint32(power_power_uint32(A2),N)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => aa_uint32_o(dvd_dvd_uint32(aa_nat_uint32(power_power_uint32(A2),M)),B2) ) ) ).

% power_le_dvd
tff(fact_5717_power__le__dvd,axiom,
    ! [A2: word_N3645301735248828278l_num1,N: nat,B2: word_N3645301735248828278l_num1,M: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A2),N)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A2),M)),B2) ) ) ).

% power_le_dvd
tff(fact_5718_power__le__dvd,axiom,
    ! [A2: nat,N: nat,B2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),B2) ) ) ).

% power_le_dvd
tff(fact_5719_power__le__dvd,axiom,
    ! [A2: int,N: nat,B2: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A2),N)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A2),M)),B2) ) ) ).

% power_le_dvd
tff(fact_5720_power__le__dvd,axiom,
    ! [A2: real,N: nat,B2: real,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),B2) ) ) ).

% power_le_dvd
tff(fact_5721_power__le__dvd,axiom,
    ! [A2: complex,N: nat,B2: complex,M: nat] :
      ( aa_complex_o(dvd_dvd_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => aa_complex_o(dvd_dvd_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),M)),B2) ) ) ).

% power_le_dvd
tff(fact_5722_power__le__dvd,axiom,
    ! [A2: code_integer,N: nat,B2: code_integer,M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),B2) ) ) ).

% power_le_dvd
tff(fact_5723_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: uint32] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_uint32_o(dvd_dvd_uint32(aa_nat_uint32(power_power_uint32(A2),M)),aa_nat_uint32(power_power_uint32(A2),N)) ) ).

% le_imp_power_dvd
tff(fact_5724_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: word_N3645301735248828278l_num1] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A2),M)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(A2),N)) ) ).

% le_imp_power_dvd
tff(fact_5725_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)) ) ).

% le_imp_power_dvd
tff(fact_5726_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),N)) ) ).

% le_imp_power_dvd
tff(fact_5727_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ) ).

% le_imp_power_dvd
tff(fact_5728_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_complex_o(dvd_dvd_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),M)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)) ) ).

% le_imp_power_dvd
tff(fact_5729_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A2),M)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ) ).

% le_imp_power_dvd
tff(fact_5730_mod__eq__dvd__iff,axiom,
    ! [A2: int,C: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)) ) ).

% mod_eq_dvd_iff
tff(fact_5731_mod__eq__dvd__iff,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),C) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(B2),C) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2)) ) ).

% mod_eq_dvd_iff
tff(fact_5732_dvd__minus__mod,axiom,
    ! [B2: nat,A2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2))) ).

% dvd_minus_mod
tff(fact_5733_dvd__minus__mod,axiom,
    ! [B2: int,A2: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2))) ).

% dvd_minus_mod
tff(fact_5734_dvd__minus__mod,axiom,
    ! [B2: code_integer,A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2))) ).

% dvd_minus_mod
tff(fact_5735_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N),M) ) ) ).

% nat_dvd_not_less
tff(fact_5736_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N) ) ) ).

% dvd_minus_self
tff(fact_5737_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_5738_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),N) ) ) ) ).

% dvd_diffD1
tff(fact_5739_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),N)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_5740_even__of__int__iff,axiom,
    ! [K: int] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_int_uint32(ring_1_of_int_uint32,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).

% even_of_int_iff
tff(fact_5741_even__of__int__iff,axiom,
    ! [K: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(ring_18347121197199848620nteger,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).

% even_of_int_iff
tff(fact_5742_even__of__int__iff,axiom,
    ! [K: int] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),ring_17408606157368542149l_num1(K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).

% even_of_int_iff
tff(fact_5743_even__of__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).

% even_of_int_iff
tff(fact_5744_even__numeral,axiom,
    ! [N: num] : aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),numera9087168376688890119uint32(aa_num_num(bit0,N))) ).

% even_numeral
tff(fact_5745_even__numeral,axiom,
    ! [N: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit0,N))) ).

% even_numeral
tff(fact_5746_even__numeral,axiom,
    ! [N: num] : aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),numera7442385471795722001l_num1(aa_num_num(bit0,N))) ).

% even_numeral
tff(fact_5747_even__numeral,axiom,
    ! [N: num] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N))) ).

% even_numeral
tff(fact_5748_even__numeral,axiom,
    ! [N: num] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).

% even_numeral
tff(fact_5749_unity__coeff__ex,axiom,
    ! [P: fun_uint32_o,L: uint32] :
      ( ? [X: uint32] : aa_uint32_o(P,aa_uint32_uint32(times_times_uint32(L),X))
    <=> ? [X: uint32] :
          ( aa_uint32_o(dvd_dvd_uint32(L),plus_plus_uint32(X,zero_zero_uint32))
          & aa_uint32_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_5750_unity__coeff__ex,axiom,
    ! [P: fun_Code_integer_o,L: code_integer] :
      ( ? [X: code_integer] : aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X))
    <=> ? [X: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(X),zero_z3403309356797280102nteger))
          & aa_Code_integer_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_5751_unity__coeff__ex,axiom,
    ! [P: fun_wo7859709073469921915num1_o,L: word_N3645301735248828278l_num1] :
      ( ? [X: word_N3645301735248828278l_num1] : aa_wor4452366487969921764num1_o(P,aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(L),X))
    <=> ? [X: word_N3645301735248828278l_num1] :
          ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(L),plus_p361126936061061375l_num1(X,zero_z3563351764282998399l_num1))
          & aa_wor4452366487969921764num1_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_5752_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X: real] : aa_real_o(P,aa_real_real(times_times_real(L),X))
    <=> ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),zero_zero_real))
          & aa_real_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_5753_unity__coeff__ex,axiom,
    ! [P: fun_rat_o,L: rat] :
      ( ? [X: rat] : aa_rat_o(P,aa_rat_rat(times_times_rat(L),X))
    <=> ? [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),plus_plus_rat(X,zero_zero_rat))
          & aa_rat_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_5754_unity__coeff__ex,axiom,
    ! [P: fun_nat_o,L: nat] :
      ( ? [X: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X))
    <=> ? [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),zero_zero_nat))
          & aa_nat_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_5755_unity__coeff__ex,axiom,
    ! [P: fun_int_o,L: int] :
      ( ? [X: int] : aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X))
    <=> ? [X: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),zero_zero_int))
          & aa_int_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_5756_unit__dvdE,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => ~ ( ( A2 != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] : B2 != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C2) ) ) ).

% unit_dvdE
tff(fact_5757_unit__dvdE,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ~ ( ( A2 != zero_zero_nat )
         => ! [C2: nat] : B2 != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2) ) ) ).

% unit_dvdE
tff(fact_5758_unit__dvdE,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ~ ( ( A2 != zero_zero_int )
         => ! [C2: int] : B2 != aa_int_int(aa_int_fun_int_int(times_times_int,A2),C2) ) ) ).

% unit_dvdE
tff(fact_5759_unit__div__eq__0__iff,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2) = zero_z3403309356797280102nteger )
      <=> ( A2 = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_5760_unit__div__eq__0__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) = zero_zero_nat )
      <=> ( A2 = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_5761_unit__div__eq__0__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = zero_zero_int )
      <=> ( A2 = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_5762_dvd__div__eq__mult,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2)
       => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),A2) = C )
        <=> ( B2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A2) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_5763_dvd__div__eq__mult,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2) = C )
        <=> ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A2) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_5764_dvd__div__eq__mult,axiom,
    ! [A2: int,B2: int,C: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2) = C )
        <=> ( B2 = aa_int_int(aa_int_fun_int_int(times_times_int,C),A2) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_5765_div__dvd__iff__mult,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),C)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_5766_div__dvd__iff__mult,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_5767_div__dvd__iff__mult,axiom,
    ! [B2: int,A2: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_5768_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),C))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),B2) ) ) ) ).

% dvd_div_iff_mult
tff(fact_5769_dvd__div__iff__mult,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),B2) ) ) ) ).

% dvd_div_iff_mult
tff(fact_5770_dvd__div__iff__mult,axiom,
    ! [C: int,B2: int,A2: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),B2) ) ) ) ).

% dvd_div_iff_mult
tff(fact_5771_dvd__div__div__eq__mult,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
           => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),A2) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(D),C) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_5772_dvd__div__div__eq__mult,axiom,
    ! [A2: nat,C: nat,B2: nat,D: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_5773_dvd__div__div__eq__mult,axiom,
    ! [A2: int,C: int,B2: int,D: int] :
      ( ( A2 != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D),C) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B2),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_5774_unit__eq__div1,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2) = C )
      <=> ( A2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2) ) ) ) ).

% unit_eq_div1
tff(fact_5775_unit__eq__div1,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) = C )
      <=> ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2) ) ) ) ).

% unit_eq_div1
tff(fact_5776_unit__eq__div1,axiom,
    ! [B2: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = C )
      <=> ( A2 = aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) ) ) ) ).

% unit_eq_div1
tff(fact_5777_unit__eq__div2,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( ( A2 = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),B2) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2) = C ) ) ) ).

% unit_eq_div2
tff(fact_5778_unit__eq__div2,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B2) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = C ) ) ) ).

% unit_eq_div2
tff(fact_5779_unit__eq__div2,axiom,
    ! [B2: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( ( A2 = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B2) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) = C ) ) ) ).

% unit_eq_div2
tff(fact_5780_div__mult__unit2,axiom,
    ! [C: code_integer,B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),C) ) ) ) ).

% div_mult_unit2
tff(fact_5781_div__mult__unit2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C) ) ) ) ).

% div_mult_unit2
tff(fact_5782_div__mult__unit2,axiom,
    ! [C: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C) ) ) ) ).

% div_mult_unit2
tff(fact_5783_unit__div__commute,axiom,
    ! [B2: code_integer,A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),C) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),C)),B2) ) ) ).

% unit_div_commute
tff(fact_5784_unit__div__commute,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C)),B2) ) ) ).

% unit_div_commute
tff(fact_5785_unit__div__commute,axiom,
    ! [B2: int,A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),C)),B2) ) ) ).

% unit_div_commute
tff(fact_5786_unit__div__mult__swap,axiom,
    ! [C: code_integer,A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)),C) ) ) ).

% unit_div_mult_swap
tff(fact_5787_unit__div__mult__swap,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C) ) ) ).

% unit_div_mult_swap
tff(fact_5788_unit__div__mult__swap,axiom,
    ! [C: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C) ) ) ).

% unit_div_mult_swap
tff(fact_5789_is__unit__div__mult2__eq,axiom,
    ! [B2: code_integer,C: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_5790_is__unit__div__mult2__eq,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_5791_is__unit__div__mult2__eq,axiom,
    ! [B2: int,C: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_5792_unit__imp__mod__eq__0,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_5793_unit__imp__mod__eq__0,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_5794_unit__imp__mod__eq__0,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_5795_is__unit__power__iff,axiom,
    ! [A2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5796_is__unit__power__iff,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A2),N)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5797_is__unit__power__iff,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_5798_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) ) ) ).

% dvd_imp_le
tff(fact_5799_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_5800_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N) ) ) ).

% dvd_mult_cancel
tff(fact_5801_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_5802_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),Q2) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_5803_real__of__nat__div,axiom,
    ! [D: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),N)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),D)) = aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(semiri5074537144036343181t_real,D)) ) ) ).

% real_of_nat_div
tff(fact_5804_even__zero,axiom,
    aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),zero_zero_uint32) ).

% even_zero
tff(fact_5805_even__zero,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_z3403309356797280102nteger) ).

% even_zero
tff(fact_5806_even__zero,axiom,
    aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),zero_z3563351764282998399l_num1) ).

% even_zero
tff(fact_5807_even__zero,axiom,
    aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_zero_nat) ).

% even_zero
tff(fact_5808_even__zero,axiom,
    aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),zero_zero_int) ).

% even_zero
tff(fact_5809_odd__even__add,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ~ aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
     => ( ~ aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),B2)
       => aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),plus_plus_uint32(A2,B2)) ) ) ).

% odd_even_add
tff(fact_5810_odd__even__add,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)) ) ) ).

% odd_even_add
tff(fact_5811_odd__even__add,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
     => ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),B2)
       => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),plus_p361126936061061375l_num1(A2,B2)) ) ) ).

% odd_even_add
tff(fact_5812_odd__even__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ) ) ).

% odd_even_add
tff(fact_5813_odd__even__add,axiom,
    ! [A2: int,B2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)) ) ) ).

% odd_even_add
tff(fact_5814_odd__one,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ).

% odd_one
tff(fact_5815_odd__one,axiom,
    ~ aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),one_one_uint32) ).

% odd_one
tff(fact_5816_odd__one,axiom,
    ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),one_on7727431528512463931l_num1) ).

% odd_one
tff(fact_5817_odd__one,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),one_one_nat) ).

% odd_one
tff(fact_5818_odd__one,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int) ).

% odd_one
tff(fact_5819_evenE,axiom,
    ! [A2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
     => ~ ! [B: uint32] : A2 != aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),B) ) ).

% evenE
tff(fact_5820_evenE,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
     => ~ ! [B: code_integer] : A2 != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ).

% evenE
tff(fact_5821_evenE,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
     => ~ ! [B: word_N3645301735248828278l_num1] : A2 != aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),B) ) ).

% evenE
tff(fact_5822_evenE,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ~ ! [B: nat] : A2 != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ).

% evenE
tff(fact_5823_evenE,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ~ ! [B: int] : A2 != aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ).

% evenE
tff(fact_5824_bit__eq__rec,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 = B2 )
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2) )
        & ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(B2),numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_5825_bit__eq__rec,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( A2 = B2 )
    <=> ( ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
        <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),B2) )
        & ( divide1791077408188789448l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = divide1791077408188789448l_num1(B2,numera7442385471795722001l_num1(aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_5826_bit__eq__rec,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( A2 = B2 )
    <=> ( ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
        <=> aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),B2) )
        & ( aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))) = aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,B2),numera9087168376688890119uint32(aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_5827_bit__eq__rec,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2) )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_5828_bit__eq__rec,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
    <=> ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2) )
        & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_5829_is__unitE,axiom,
    ! [A2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
     => ~ ( ( A2 != zero_z3403309356797280102nteger )
         => ! [B: code_integer] :
              ( ( B != zero_z3403309356797280102nteger )
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
               => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),A2) = B )
                 => ( ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),B) = A2 )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B) = one_one_Code_integer )
                     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(C),A2) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_5830_is__unitE,axiom,
    ! [A2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ~ ( ( A2 != zero_zero_nat )
         => ! [B: nat] :
              ( ( B != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A2) = B )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) = A2 )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A2) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_5831_is__unitE,axiom,
    ! [A2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ~ ( ( A2 != zero_zero_int )
         => ! [B: int] :
              ( ( B != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A2) = B )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) = A2 )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),B) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A2) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_5832_is__unit__div__mult__cancel__left,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),B2) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_5833_is__unit__div__mult__cancel__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_5834_is__unit__div__mult__cancel__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B2) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_5835_is__unit__div__mult__cancel__right,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
       => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),A2)) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(one_one_Code_integer),B2) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_5836_is__unit__div__mult__cancel__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_5837_is__unit__div__mult__cancel__right,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(times_times_int,B2),A2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B2) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_5838_odd__numeral,axiom,
    ! [N: num] : ~ aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),numera9087168376688890119uint32(aa_num_num(bit1,N))) ).

% odd_numeral
tff(fact_5839_odd__numeral,axiom,
    ! [N: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit1,N))) ).

% odd_numeral
tff(fact_5840_odd__numeral,axiom,
    ! [N: num] : ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),numera7442385471795722001l_num1(aa_num_num(bit1,N))) ).

% odd_numeral
tff(fact_5841_odd__numeral,axiom,
    ! [N: num] : ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,N))) ).

% odd_numeral
tff(fact_5842_odd__numeral,axiom,
    ! [N: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).

% odd_numeral
tff(fact_5843_dvd__power__iff,axiom,
    ! [Xb: nat,M: nat,N: nat] :
      ( ( Xb != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),N))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ) ).

% dvd_power_iff
tff(fact_5844_dvd__power__iff,axiom,
    ! [Xb: int,M: nat,N: nat] :
      ( ( Xb != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),M)),aa_nat_int(power_power_int(Xb),N))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ) ).

% dvd_power_iff
tff(fact_5845_dvd__power__iff,axiom,
    ! [Xb: code_integer,M: nat,N: nat] :
      ( ( Xb != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),M)),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),N))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ) ).

% dvd_power_iff
tff(fact_5846_dvd__power,axiom,
    ! [N: nat,Xb: word_N3645301735248828278l_num1] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( Xb = one_on7727431528512463931l_num1 ) )
     => aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Xb),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(Xb),N)) ) ).

% dvd_power
tff(fact_5847_dvd__power,axiom,
    ! [N: nat,Xb: uint32] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( Xb = one_one_uint32 ) )
     => aa_uint32_o(dvd_dvd_uint32(Xb),aa_nat_uint32(power_power_uint32(Xb),N)) ) ).

% dvd_power
tff(fact_5848_dvd__power,axiom,
    ! [N: nat,Xb: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( Xb = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),aa_nat_rat(power_power_rat(Xb),N)) ) ).

% dvd_power
tff(fact_5849_dvd__power,axiom,
    ! [N: nat,Xb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( Xb = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),N)) ) ).

% dvd_power
tff(fact_5850_dvd__power,axiom,
    ! [N: nat,Xb: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( Xb = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_nat_int(power_power_int(Xb),N)) ) ).

% dvd_power
tff(fact_5851_dvd__power,axiom,
    ! [N: nat,Xb: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( Xb = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)) ) ).

% dvd_power
tff(fact_5852_dvd__power,axiom,
    ! [N: nat,Xb: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( Xb = one_one_complex ) )
     => aa_complex_o(dvd_dvd_complex(Xb),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),N)) ) ).

% dvd_power
tff(fact_5853_dvd__power,axiom,
    ! [N: nat,Xb: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( Xb = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),N)) ) ).

% dvd_power
tff(fact_5854_refines__If,axiom,
    ! [B2: $o,Tb: heap_T2636463487746394924on_nat,T2: heap_T2636463487746394924on_nat,E2: heap_T2636463487746394924on_nat,E3: heap_T2636463487746394924on_nat] :
      ( ( (B2)
       => refine7594492741263601813on_nat(Tb,T2) )
     => ( ( ~ (B2)
         => refine7594492741263601813on_nat(E2,E3) )
       => refine7594492741263601813on_nat($ite((B2),Tb,E2),
            $ite((B2),T2,E3)) ) ) ).

% refines_If
tff(fact_5855_refines__If,axiom,
    ! [B2: $o,Tb: heap_Time_Heap_o,T2: heap_Time_Heap_o,E2: heap_Time_Heap_o,E3: heap_Time_Heap_o] :
      ( ( (B2)
       => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(Tb),T2) )
     => ( ( ~ (B2)
         => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(E2),E3) )
       => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o($ite((B2),Tb,E2)),
            $ite((B2),T2,E3)) ) ) ).

% refines_If
tff(fact_5856_refines__If,axiom,
    ! [B2: $o,Tb: heap_T8145700208782473153_VEBTi,T2: heap_T8145700208782473153_VEBTi,E2: heap_T8145700208782473153_VEBTi,E3: heap_T8145700208782473153_VEBTi] :
      ( ( (B2)
       => refine5565527176597971370_VEBTi(Tb,T2) )
     => ( ( ~ (B2)
         => refine5565527176597971370_VEBTi(E2,E3) )
       => refine5565527176597971370_VEBTi($ite((B2),Tb,E2),
            $ite((B2),T2,E3)) ) ) ).

% refines_If
tff(fact_5857_refines__refl,axiom,
    ! [P2: heap_T2636463487746394924on_nat] : refine7594492741263601813on_nat(P2,P2) ).

% refines_refl
tff(fact_5858_refines__refl,axiom,
    ! [P2: heap_Time_Heap_o] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(P2),P2) ).

% refines_refl
tff(fact_5859_refines__refl,axiom,
    ! [P2: heap_T8145700208782473153_VEBTi] : refine5565527176597971370_VEBTi(P2,P2) ).

% refines_refl
tff(fact_5860_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% even_even_mod_4_iff
tff(fact_5861_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),M)
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_5862_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M)),M)
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_5863_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% power_dvd_imp_le
tff(fact_5864_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q2),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),M))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),Q2))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),M)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_5865_mod__nat__eqI,axiom,
    ! [R: nat,N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N) = R ) ) ) ) ).

% mod_nat_eqI
tff(fact_5866_diff__mod__le,axiom,
    ! [A2: nat,D: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),D)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,D),B2)) ) ) ).

% diff_mod_le
tff(fact_5867_even__two__times__div__two,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))) = A2 ) ) ).

% even_two_times_div_two
tff(fact_5868_even__two__times__div__two,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = A2 ) ) ).

% even_two_times_div_two
tff(fact_5869_even__two__times__div__two,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = A2 ) ) ).

% even_two_times_div_two
tff(fact_5870_even__iff__mod__2__eq__zero,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
    <=> ( modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = zero_z3563351764282998399l_num1 ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5871_even__iff__mod__2__eq__zero,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5872_even__iff__mod__2__eq__zero,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5873_even__iff__mod__2__eq__zero,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5874_even__iff__mod__2__eq__zero,axiom,
    ! [A2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
    <=> ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))) = zero_zero_uint32 ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5875_odd__iff__mod__2__eq__one,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
    <=> ( modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = one_on7727431528512463931l_num1 ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5876_odd__iff__mod__2__eq__one,axiom,
    ! [A2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5877_odd__iff__mod__2__eq__one,axiom,
    ! [A2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5878_odd__iff__mod__2__eq__one,axiom,
    ! [A2: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5879_odd__iff__mod__2__eq__one,axiom,
    ! [A2: uint32] :
      ( ~ aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
    <=> ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))) = one_one_uint32 ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5880_power__mono__odd,axiom,
    ! [N: nat,A2: real,B2: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)) ) ) ).

% power_mono_odd
tff(fact_5881_power__mono__odd,axiom,
    ! [N: nat,A2: code_integer,B2: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N)) ) ) ).

% power_mono_odd
tff(fact_5882_power__mono__odd,axiom,
    ! [N: nat,A2: rat,B2: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B2),N)) ) ) ).

% power_mono_odd
tff(fact_5883_power__mono__odd,axiom,
    ! [N: nat,A2: int,B2: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(B2),N)) ) ) ).

% power_mono_odd
tff(fact_5884_odd__pos,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% odd_pos
tff(fact_5885_even__set__bit__iff,axiom,
    ! [M: nat,A2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),bit_se6647067497041451410uint32(M,A2))
    <=> ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5886_even__set__bit__iff,axiom,
    ! [M: nat,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se2793503036327961859nteger(M),A2))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5887_even__set__bit__iff,axiom,
    ! [M: nat,A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),bit_se4894374433684937756l_num1(M,A2))
    <=> ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5888_even__set__bit__iff,axiom,
    ! [M: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(M),A2))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5889_even__set__bit__iff,axiom,
    ! [M: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5890_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),N))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% dvd_power_iff_le
tff(fact_5891_oddE,axiom,
    ! [A2: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
     => ~ ! [B: code_integer] : A2 != aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)),one_one_Code_integer) ) ).

% oddE
tff(fact_5892_oddE,axiom,
    ! [A2: uint32] :
      ( ~ aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
     => ~ ! [B: uint32] : A2 != plus_plus_uint32(aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),B),one_one_uint32) ) ).

% oddE
tff(fact_5893_oddE,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
     => ~ ! [B: word_N3645301735248828278l_num1] : A2 != plus_p361126936061061375l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),B),one_on7727431528512463931l_num1) ) ).

% oddE
tff(fact_5894_oddE,axiom,
    ! [A2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ~ ! [B: nat] : A2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)),one_one_nat) ) ).

% oddE
tff(fact_5895_oddE,axiom,
    ! [A2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ~ ! [B: int] : A2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),one_one_int) ) ).

% oddE
tff(fact_5896_mod2__eq__if,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))) = $ite(aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2),zero_z3563351764282998399l_num1,one_on7727431528512463931l_num1) ).

% mod2_eq_if
tff(fact_5897_mod2__eq__if,axiom,
    ! [A2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_5898_mod2__eq__if,axiom,
    ! [A2: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_5899_mod2__eq__if,axiom,
    ! [A2: code_integer] :
      aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_5900_mod2__eq__if,axiom,
    ! [A2: uint32] :
      aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))) = $ite(aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2),zero_zero_uint32,one_one_uint32) ).

% mod2_eq_if
tff(fact_5901_parity__cases,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
       => ( modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))) != zero_z3563351764282998399l_num1 ) )
     => ~ ( ~ aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
         => ( modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))) != one_on7727431528512463931l_num1 ) ) ) ).

% parity_cases
tff(fact_5902_parity__cases,axiom,
    ! [A2: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat ) )
     => ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_5903_parity__cases,axiom,
    ! [A2: int] :
      ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_5904_parity__cases,axiom,
    ! [A2: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
       => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
         => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_5905_parity__cases,axiom,
    ! [A2: uint32] :
      ( ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
       => ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))) != zero_zero_uint32 ) )
     => ~ ( ~ aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
         => ( aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))) != one_one_uint32 ) ) ) ).

% parity_cases
tff(fact_5906_zero__le__power__eq,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ) ) ).

% zero_le_power_eq
tff(fact_5907_zero__le__power__eq,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2) ) ) ) ).

% zero_le_power_eq
tff(fact_5908_zero__le__power__eq,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ) ) ).

% zero_le_power_eq
tff(fact_5909_zero__le__power__eq,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),N))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ) ) ).

% zero_le_power_eq
tff(fact_5910_zero__le__odd__power,axiom,
    ! [N: nat,A2: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ) ).

% zero_le_odd_power
tff(fact_5911_zero__le__odd__power,axiom,
    ! [N: nat,A2: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2) ) ) ).

% zero_le_odd_power
tff(fact_5912_zero__le__odd__power,axiom,
    ! [N: nat,A2: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),N))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ) ).

% zero_le_odd_power
tff(fact_5913_zero__le__odd__power,axiom,
    ! [N: nat,A2: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),N))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ) ).

% zero_le_odd_power
tff(fact_5914_zero__le__even__power,axiom,
    ! [N: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ) ).

% zero_le_even_power
tff(fact_5915_zero__le__even__power,axiom,
    ! [N: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ) ).

% zero_le_even_power
tff(fact_5916_zero__le__even__power,axiom,
    ! [N: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),N)) ) ).

% zero_le_even_power
tff(fact_5917_zero__le__even__power,axiom,
    ! [N: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),N)) ) ).

% zero_le_even_power
tff(fact_5918_divides__aux__def,axiom,
    ! [Qr: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,Qr)
    <=> ( aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Qr) = zero_zero_nat ) ) ).

% divides_aux_def
tff(fact_5919_divides__aux__def,axiom,
    ! [Qr: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,Qr)
    <=> ( aa_Pro6799258434465214121nt_int(product_snd_int_int,Qr) = zero_zero_int ) ) ).

% divides_aux_def
tff(fact_5920_zero__less__power__eq,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))
    <=> ( ( N = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & ( A2 != zero_z3403309356797280102nteger ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2) ) ) ) ).

% zero_less_power_eq
tff(fact_5921_zero__less__power__eq,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))
    <=> ( ( N = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & ( A2 != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2) ) ) ) ).

% zero_less_power_eq
tff(fact_5922_zero__less__power__eq,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),N))
    <=> ( ( N = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & ( A2 != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ) ) ).

% zero_less_power_eq
tff(fact_5923_zero__less__power__eq,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A2),N))
    <=> ( ( N = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & ( A2 != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2) ) ) ) ).

% zero_less_power_eq
tff(fact_5924_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)),one_one_Code_integer)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% even_mask_div_iff'
tff(fact_5925_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% even_mask_div_iff'
tff(fact_5926_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% even_mask_div_iff'
tff(fact_5927_power__le__zero__eq,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( A2 = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5928_power__le__zero__eq,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),zero_z3403309356797280102nteger)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( A2 = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5929_power__le__zero__eq,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( A2 = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5930_power__le__zero__eq,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),N)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( A2 = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5931_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% even_mod_4_div_2
tff(fact_5932_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M)),one_one_Code_integer)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% even_mask_div_iff
tff(fact_5933_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),divide1791077408188789448l_num1(minus_4019991460397169231l_num1(aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),M),one_on7727431528512463931l_num1),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N)))
    <=> ( ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N) = zero_z3563351764282998399l_num1 )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% even_mask_div_iff
tff(fact_5934_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,minus_minus_uint32(aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),M),one_one_uint32)),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),N)))
    <=> ( ( aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),N) = zero_zero_uint32 )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% even_mask_div_iff
tff(fact_5935_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% even_mask_div_iff
tff(fact_5936_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ) ).

% even_mask_div_iff
tff(fact_5937_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% odd_mod_4_div_2
tff(fact_5938_refines__bind,axiom,
    ! [M: heap_Time_Heap_nat,M5: heap_Time_Heap_nat,Fa: fun_na2343073368905879023Heap_o,F: fun_na2343073368905879023Heap_o] :
      ( refine1365783493865988805es_nat(M,M5)
     => ( ! [X2: nat] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_nat2910048432523430448Heap_o(Fa,X2)),aa_nat2910048432523430448Heap_o(F,X2))
       => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(heap_Time_bind_nat_o(M,Fa)),heap_Time_bind_nat_o(M5,F)) ) ) ).

% refines_bind
tff(fact_5939_refines__bind,axiom,
    ! [M: heap_Time_Heap_nat,M5: heap_Time_Heap_nat,Fa: fun_na7848093160509028170_VEBTi,F: fun_na7848093160509028170_VEBTi] :
      ( refine1365783493865988805es_nat(M,M5)
     => ( ! [X2: nat] : refine5565527176597971370_VEBTi(aa_nat5408703056993926195_VEBTi(Fa,X2),aa_nat5408703056993926195_VEBTi(F,X2))
       => refine5565527176597971370_VEBTi(heap_T844888390831797134_VEBTi(M,Fa),heap_T844888390831797134_VEBTi(M5,F)) ) ) ).

% refines_bind
tff(fact_5940_refines__bind,axiom,
    ! [M: heap_Time_Heap_o,M5: heap_Time_Heap_o,Fa: fun_o_8325599348076409421Heap_o,F: fun_o_8325599348076409421Heap_o] :
      ( aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(M),M5)
     => ( ! [X2: $o] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_o_H7399033255965945470Heap_o(Fa,(X2))),aa_o_H7399033255965945470Heap_o(F,(X2)))
       => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(heap_Time_bind_o_o(M,Fa)),heap_Time_bind_o_o(M5,F)) ) ) ).

% refines_bind
tff(fact_5941_refines__bind,axiom,
    ! [M: heap_Time_Heap_o,M5: heap_Time_Heap_o,Fa: fun_o_2238389867833408620_VEBTi,F: fun_o_2238389867833408620_VEBTi] :
      ( aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(M),M5)
     => ( ! [X2: $o] : refine5565527176597971370_VEBTi(aa_o_H7577119287286566245_VEBTi(Fa,(X2)),aa_o_H7577119287286566245_VEBTi(F,(X2)))
       => refine5565527176597971370_VEBTi(heap_T5998771940306268294_VEBTi(M,Fa),heap_T5998771940306268294_VEBTi(M5,F)) ) ) ).

% refines_bind
tff(fact_5942_refines__bind,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,M5: heap_T8145700208782473153_VEBTi,Fa: fun_VE7038733415900436564Heap_o,F: fun_VE7038733415900436564Heap_o] :
      ( refine5565527176597971370_VEBTi(M,M5)
     => ( ! [X2: vEBT_VEBTi] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_VEB8194921341634505877Heap_o(Fa,X2)),aa_VEB8194921341634505877Heap_o(F,X2))
       => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(heap_T3040810144269856602EBTi_o(M,Fa)),heap_T3040810144269856602EBTi_o(M5,F)) ) ) ).

% refines_bind
tff(fact_5943_refines__bind,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,M5: heap_T8145700208782473153_VEBTi,Fa: fun_VE4765431424076933413_VEBTi,F: fun_VE4765431424076933413_VEBTi] :
      ( refine5565527176597971370_VEBTi(M,M5)
     => ( ! [X2: vEBT_VEBTi] : refine5565527176597971370_VEBTi(aa_VEB836174949890266638_VEBTi(Fa,X2),aa_VEB836174949890266638_VEBTi(F,X2))
       => refine5565527176597971370_VEBTi(heap_T1006145433769338483_VEBTi(M,Fa),heap_T1006145433769338483_VEBTi(M5,F)) ) ) ).

% refines_bind
tff(fact_5944_refines__bind,axiom,
    ! [M: heap_Time_Heap_nat,M5: heap_Time_Heap_nat,Fa: fun_na5556145776640407093on_nat,F: fun_na5556145776640407093on_nat] :
      ( refine1365783493865988805es_nat(M,M5)
     => ( ! [X2: nat] : refine7594492741263601813on_nat(aa_nat947733127081326238on_nat(Fa,X2),aa_nat947733127081326238on_nat(F,X2))
       => refine7594492741263601813on_nat(heap_T8222160169144143993on_nat(M,Fa),heap_T8222160169144143993on_nat(M5,F)) ) ) ).

% refines_bind
tff(fact_5945_refines__bind,axiom,
    ! [M: heap_T4980287057938770641_VEBTi,M5: heap_T4980287057938770641_VEBTi,Fa: fun_li7499208429157053461_VEBTi,F: fun_li7499208429157053461_VEBTi] :
      ( refine3700189196150522554_VEBTi(M,M5)
     => ( ! [X2: list_VEBT_VEBTi] : refine5565527176597971370_VEBTi(aa_lis1839255523752068350_VEBTi(Fa,X2),aa_lis1839255523752068350_VEBTi(F,X2))
       => refine5565527176597971370_VEBTi(heap_T5877712393672139267_VEBTi(M,Fa),heap_T5877712393672139267_VEBTi(M5,F)) ) ) ).

% refines_bind
tff(fact_5946_refines__bind,axiom,
    ! [M: heap_T2636463487746394924on_nat,M5: heap_T2636463487746394924on_nat,Fa: fun_op3694375823608013759Heap_o,F: fun_op3694375823608013759Heap_o] :
      ( refine7594492741263601813on_nat(M,M5)
     => ( ! [X2: option_nat] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_opt6753654531493786624Heap_o(Fa,X2)),aa_opt6753654531493786624Heap_o(F,X2))
       => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(heap_T6471384023045698863_nat_o(M,Fa)),heap_T6471384023045698863_nat_o(M5,F)) ) ) ).

% refines_bind
tff(fact_5947_refines__bind,axiom,
    ! [M: heap_T2636463487746394924on_nat,M5: heap_T2636463487746394924on_nat,Fa: fun_op1509349959255590266_VEBTi,F: fun_op1509349959255590266_VEBTi] :
      ( refine7594492741263601813on_nat(M,M5)
     => ( ! [X2: option_nat] : refine5565527176597971370_VEBTi(aa_opt6483592524530923107_VEBTi(Fa,X2),aa_opt6483592524530923107_VEBTi(F,X2))
       => refine5565527176597971370_VEBTi(heap_T5661892481228163294_VEBTi(M,Fa),heap_T5661892481228163294_VEBTi(M5,F)) ) ) ).

% refines_bind
tff(fact_5948_Bernoulli__inequality__even,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),N)) ) ).

% Bernoulli_inequality_even
tff(fact_5949_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [N: nat] :
      aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
tff(fact_5950_even__mult__exp__div__exp__iff,axiom,
    ! [A2: code_integer,M: nat,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),M))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5951_even__mult__exp__div__exp__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,M: nat,N: nat] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),divide1791077408188789448l_num1(aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(A2),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),M)),aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
        | ( aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),N) = zero_z3563351764282998399l_num1 )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
          & aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),divide1791077408188789448l_num1(A2,aa_nat5599461682457136004l_num1(power_2184487114949457152l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5952_even__mult__exp__div__exp__iff,axiom,
    ! [A2: uint32,M: nat,N: nat] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,aa_uint32_uint32(times_times_uint32(A2),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),M))),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
        | ( aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),N) = zero_zero_uint32 )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
          & aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),aa_nat_uint32(power_power_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5953_even__mult__exp__div__exp__iff,axiom,
    ! [A2: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5954_even__mult__exp__div__exp__iff,axiom,
    ! [A2: int,M: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5955_refines__option,axiom,
    ! [A2: option4927543243414619207at_nat,A4: option4927543243414619207at_nat,M1: heap_Time_Heap_o,M12: heap_Time_Heap_o,M22: fun_Pr1329037631615965750Heap_o,M23: fun_Pr1329037631615965750Heap_o] :
      ( ( A2 = A4 )
     => ( aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(M1),M12)
       => ( ! [X2: product_prod_nat_nat] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(aa_Pro691548381575413351Heap_o(M22,X2)),aa_Pro691548381575413351Heap_o(M23,X2))
         => aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(case_o1442776274061689234at_nat(M1,M22,A2)),case_o1442776274061689234at_nat(M12,M23,A4)) ) ) ) ).

% refines_option
tff(fact_5956_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_nat_nat(suc,zero_zero_nat) ) )
         => ~ ! [N3: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_5957_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% VEBT_internal.Tb'.simps(3)
tff(fact_5958_VEBT__internal_OTb_H_Oelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
         => ~ ! [N3: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
tff(fact_5959_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [N: nat] :
      aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% VEBT_internal.Tb.simps(3)
tff(fact_5960_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
tff(fact_5961_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] :
      aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $let(
            half: nat,
            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_5962_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] :
      aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $let(
            half: nat,
            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_5963_VEBT__internal_OTb_Oelims,axiom,
    ! [Xb: nat,Y: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ) )
         => ~ ! [N3: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
               => ( Y != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
tff(fact_5964_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,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $let(
                          half: nat,
                          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_5965_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,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $let(
                          half: nat,
                          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_5966_pow__divides__pow__iff,axiom,
    ! [N: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2) ) ) ).

% pow_divides_pow_iff
tff(fact_5967_pow__divides__pow__iff,axiom,
    ! [N: nat,A2: int,B2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(B2),N))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ) ).

% pow_divides_pow_iff
tff(fact_5968_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_jd(L),Qr) ).

% divmod_step_integer_def
tff(fact_5969_artanh__def,axiom,
    ! [Xb: real] : aa_real_real(artanh_real,Xb) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(minus_minus_real(one_one_real),Xb)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_5970_div2__even__ext__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y) )
       => ( Xb = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_5971_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ? [D2: nat,X2: nat,Y2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A2)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B2)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y2)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_5972_flip__bit__Suc,axiom,
    ! [N: nat,A2: word_N3645301735248828278l_num1] : bit_se4491814353640558621l_num1(aa_nat_nat(suc,N),A2) = plus_p361126936061061375l_num1(modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),bit_se4491814353640558621l_num1(N,divide1791077408188789448l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5973_flip__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] : aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(aa_nat_nat(suc,N)),A2) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5974_flip__bit__Suc,axiom,
    ! [N: nat,A2: uint32] : bit_se7025624438249859091uint32(aa_nat_nat(suc,N),A2) = plus_plus_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),bit_se7025624438249859091uint32(N,aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5975_flip__bit__Suc,axiom,
    ! [N: nat,A2: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,N)),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5976_flip__bit__Suc,axiom,
    ! [N: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,N)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5977_ln__inj__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( ( aa_real_real(ln_ln_real,Xb) = aa_real_real(ln_ln_real,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_5978_ln__less__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_5979_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2159334234014336723it_int(N),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_5980_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2159334234014336723it_int(N),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% flip_bit_negative_int_iff
tff(fact_5981_ln__le__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_5982_ln__eq__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( ( aa_real_real(ln_ln_real,Xb) = zero_zero_real )
      <=> ( Xb = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_5983_ln__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% ln_gt_zero_iff
tff(fact_5984_ln__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% ln_less_zero_iff
tff(fact_5985_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_5986_ln__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% ln_le_zero_iff
tff(fact_5987_ln__ge__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% ln_ge_zero_iff
tff(fact_5988_ln__less__self,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),Xb) ) ).

% ln_less_self
tff(fact_5989_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),N)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),N)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,N),M)
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_5990_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,M),N)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,N),M) ) ) ).

% zdvd_not_zless
tff(fact_5991_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),N))
     => ( ( K != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),N) ) ) ).

% zdvd_mult_cancel
tff(fact_5992_zdvd__mono,axiom,
    ! [K: int,M: int,Tb: int] :
      ( ( K != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Tb)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),M)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Tb)) ) ) ).

% zdvd_mono
tff(fact_5993_ln__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),Xb) ) ).

% ln_bound
tff(fact_5994_ln__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb)) ) ).

% ln_gt_zero
tff(fact_5995_ln__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real) ) ) ).

% ln_less_zero
tff(fact_5996_ln__gt__zero__imp__gt__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_5997_ln__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb)) ) ).

% ln_ge_zero
tff(fact_5998_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),N) ) ) ).

% zdvd_imp_le
tff(fact_5999_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_6000_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_6001_real__of__int__div,axiom,
    ! [D: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),N)
     => ( aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),D)) = aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,N)),aa_int_real(ring_1_of_int_real,D)) ) ) ).

% real_of_int_div
tff(fact_6002_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
tff(fact_6003_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
tff(fact_6004_prod__decode__aux_Ocases,axiom,
    ! [Xb: product_prod_nat_nat] :
      ~ ! [K2: nat,M3: nat] : Xb != aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,K2),M3) ).

% prod_decode_aux.cases
tff(fact_6005_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% times_integer_code(1)
tff(fact_6006_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),L) = zero_z3403309356797280102nteger ).

% times_integer_code(2)
tff(fact_6007_less__eq__integer__code_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_eq_integer_code(1)
tff(fact_6008_ln__ge__zero__imp__ge__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_6009_ln__add__one__self__le__self,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb) ) ).

% ln_add_one_self_le_self
tff(fact_6010_ln__mult,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(times_times_real(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_6011_ln__eq__minus__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( ( aa_real_real(ln_ln_real,Xb) = aa_real_real(minus_minus_real(Xb),one_one_real) )
       => ( Xb = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_6012_ln__div,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(divide_divide_real(Xb),Y)) = aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_6013_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( ( L = zero_zero_int )
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L) ) ) ).

% mod_int_pos_iff
tff(fact_6014_int__div__sub__1,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),M)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,N),one_one_int)),M) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),N),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,N),M)),one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,N),M)) ) ) ).

% int_div_sub_1
tff(fact_6015_bset_I9_J,axiom,
    ! [D: int,D4: int,B5: set_int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B5)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Tb))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)),Tb)) ) ) ) ).

% bset(9)
tff(fact_6016_bset_I10_J,axiom,
    ! [D: int,D4: int,B5: set_int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B5)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Tb))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)),Tb)) ) ) ) ).

% bset(10)
tff(fact_6017_aset_I9_J,axiom,
    ! [D: int,D4: int,A3: set_int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A3)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Tb))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)),Tb)) ) ) ) ).

% aset(9)
tff(fact_6018_aset_I10_J,axiom,
    ! [D: int,D4: int,A3: set_int,Tb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A3)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),Tb))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)),Tb)) ) ) ) ).

% aset(10)
tff(fact_6019_ln__2__less__1,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,numeral_numeral_real(aa_num_num(bit0,one)))),one_one_real) ).

% ln_2_less_1
tff(fact_6020_ln__le__minus__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(minus_minus_real(Xb),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_6021_ln__diff__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y))),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Xb),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_6022_ln__realpow,axiom,
    ! [Xb: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_realpow
tff(fact_6023_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ) ).

% even_diff_iff
tff(fact_6024_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
         => ( ( B2 != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
             => ( aa_real_real(log(A2),Xb) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B2)),aa_real_real(ln_ln_real,A2))),aa_real_real(log(B2),Xb)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_6025_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ~ ! [N3: nat] : Xb != aa_nat_nat(suc,N3) ) ).

% list_decode.cases
tff(fact_6026_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,A2: nat,B2: nat] :
      ( ! [A: nat,B: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P,A),B)
        <=> aa_nat_o(aa_nat_fun_nat_o(P,B),A) )
     => ( ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A),zero_zero_nat)
       => ( ! [A: nat,B: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,A),B)
             => aa_nat_o(aa_nat_fun_nat_o(P,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A2),B2) ) ) ) ).

% Euclid_induct
tff(fact_6027_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),zero_zero_nat) ).

% gcd_nat.extremum
tff(fact_6028_gcd__nat_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A2)
        & ( zero_zero_nat != A2 ) ) ).

% gcd_nat.extremum_strict
tff(fact_6029_gcd__nat_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A2)
    <=> ( A2 = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_6030_gcd__nat_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),zero_zero_nat)
        & ( A2 != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_6031_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A2)
     => ( A2 = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_6032_less__integer__code_I1_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_integer_code(1)
tff(fact_6033_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(K),zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_6034_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(zero_z3403309356797280102nteger),L) = L ).

% plus_integer_code(2)
tff(fact_6035_eme1p,axiom,
    ! [N: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)),D) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),D)) ) ) ) ) ).

% eme1p
tff(fact_6036_emep1,axiom,
    ! [N: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int)),D) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,N),D)),one_one_int) ) ) ) ) ).

% emep1
tff(fact_6037_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(K),zero_z3403309356797280102nteger) = K ).

% minus_integer_code(1)
tff(fact_6038_even__flip__bit__iff,axiom,
    ! [M: nat,A2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),bit_se7025624438249859091uint32(M,A2))
    <=> ~ ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_6039_even__flip__bit__iff,axiom,
    ! [M: nat,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se1345352211410354436nteger(M),A2))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_6040_even__flip__bit__iff,axiom,
    ! [M: nat,A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),bit_se4491814353640558621l_num1(M,A2))
    <=> ~ ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_6041_even__flip__bit__iff,axiom,
    ! [M: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(M),A2))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_6042_even__flip__bit__iff,axiom,
    ! [M: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A2))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_6043_ln__one__plus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_6044_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M) ) ) ).

% dvd_pos_nat
tff(fact_6045_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M: nat,N: nat] :
      ( ! [M3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M3),zero_zero_nat)
     => ( ! [M3: nat,N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3)
           => ( aa_nat_o(aa_nat_fun_nat_o(P,N3),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M3),N3))
             => aa_nat_o(aa_nat_fun_nat_o(P,M3),N3) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P,M),N) ) ) ).

% gcd_nat_induct
tff(fact_6046_VEBT__internal_Ovebt__memberi_H_Osimps,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(vEBT_V854960066525838166emberi,Tb),Tib),Xb) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_nat1227502447850154924Heap_o(aTP_Lamm_nq(Tb),Xb),aTP_Lamm_nr(Xb)),Tib) ).

% VEBT_internal.vebt_memberi'.simps
tff(fact_6047_vebt__memberi_Osimps,axiom,
    ! [Tb: vEBT_VEBTi,Xb: nat] : aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(vEBT_vebt_memberi,Tb),Xb) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aTP_Lamm_nx(Xb),aTP_Lamm_nr(Xb)),Tb) ).

% vebt_memberi.simps
tff(fact_6048_VEBT__internal_Ovebt__memberi_H_Oraw__induct,axiom,
    ! [P: fun_VE6326492262185073320_nat_o,Xaa: produc3960855945107176009Ti_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: $o,N: nat] :
      ( ! [Vebt_memberi: fun_VE8181444127144381288Heap_o] :
          ( ! [A5: vEBT_VEBT,B4: vEBT_VEBTi,Ba: nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R2: $o,N4: nat] :
              ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(Vebt_memberi,A5),B4),Ba),H3,H4),(R2)),N4)
             => aa_nat_o(aa_o_fun_nat_o(aa_hea6792647905190977245_nat_o(aa_hea8977266003532396635_nat_o(aa_nat1261834975400290800_nat_o(aa_VEB7158420292131534498_nat_o(aa_VEB1179214148959353945_nat_o(P,A5),B4),Ba),H3),H4),(R2)),N4) )
         => ! [T: vEBT_VEBT,Ti2: vEBT_VEBTi,X2: nat,Ta: heap_e7401611519738050253t_unit,Tia: heap_e7401611519738050253t_unit,Xa2: $o,N3: nat] :
              ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_nat1227502447850154924Heap_o(aa_VEB6636772557416003555Heap_o(aTP_Lamm_ol(Vebt_memberi),T),X2),aTP_Lamm_nr(X2)),Ti2),Ta,Tia),(Xa2)),N3)
             => aa_nat_o(aa_o_fun_nat_o(aa_hea6792647905190977245_nat_o(aa_hea8977266003532396635_nat_o(aa_nat1261834975400290800_nat_o(aa_VEB7158420292131534498_nat_o(aa_VEB1179214148959353945_nat_o(P,T),Ti2),X2),Ta),Tia),(Xa2)),N3) ) )
     => ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_Pro6173831900156065711Heap_o(produc5872130906356439992Heap_o(produc844971584604743726Heap_o(vEBT_V854960066525838166emberi)),Xaa),H,H2),(R)),N)
       => aa_nat_o(aa_o_fun_nat_o(aa_hea6792647905190977245_nat_o(aa_hea8977266003532396635_nat_o(produc4720546487998029688_nat_o(produc8632027548060743022_nat_o(P),Xaa),H),H2),(R)),N) ) ) ).

% VEBT_internal.vebt_memberi'.raw_induct
tff(fact_6049_bezw_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Xaa)))))) ) ) ).

% bezw.elims
tff(fact_6050_bezw_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      bezw(Xb,Y) = $ite(Y = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y)))))) ).

% bezw.simps
tff(fact_6051_unset__bit__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] : bit_se5331074070815623765l_num1(zero_zero_nat,A2) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),divide1791077408188789448l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_6052_unset__bit__0,axiom,
    ! [A2: uint32] : bit_se4315839071623982667uint32(zero_zero_nat,A2) = aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_6053_unset__bit__0,axiom,
    ! [A2: int] : aa_int_int(bit_se4203085406695923979it_int(zero_zero_nat),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_6054_unset__bit__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_6055_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se4203085406695923979it_int(N),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_6056_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se4203085406695923979it_int(N),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% unset_bit_negative_int_iff
tff(fact_6057_bezw__0,axiom,
    ! [Xb: nat] : bezw(Xb,zero_zero_nat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ).

% bezw_0
tff(fact_6058_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se4203085406695923979it_int(N),K)),K) ).

% unset_bit_less_eq
tff(fact_6059_time__array__len,axiom,
    ! [P2: array_VEBT_VEBTi,H: heap_e7401611519738050253t_unit] : time_time_nat(array_len_VEBT_VEBTi(P2),H) = one_one_nat ).

% time_array_len
tff(fact_6060_TBOUND__len,axiom,
    ! [Xs: array_VEBT_VEBTi] : aa_nat_o(time_TBOUND_nat(array_len_VEBT_VEBTi(Xs)),one_one_nat) ).

% TBOUND_len
tff(fact_6061_vebt__memberi_Oraw__induct,axiom,
    ! [P: fun_VE8999033711980490209_nat_o,Xaa: produc3881548065746020326Ti_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: $o,N: nat] :
      ( ! [Vebt_memberi2: fun_VE7134199479003659425Heap_o] :
          ( ! [A5: vEBT_VEBTi,B4: nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R2: $o,N4: nat] :
              ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(Vebt_memberi2,A5),B4),H3,H4),(R2)),N4)
             => aa_nat_o(aa_o_fun_nat_o(aa_hea6792647905190977245_nat_o(aa_hea8977266003532396635_nat_o(aa_nat1261834975400290800_nat_o(aa_VEB7158420292131534498_nat_o(P,A5),B4),H3),H4),(R2)),N4) )
         => ! [T: vEBT_VEBTi,X2: nat,Ta: heap_e7401611519738050253t_unit,Xa2: heap_e7401611519738050253t_unit,R3: $o,N3: nat] :
              ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_nat1227502447850154924Heap_o(aTP_Lamm_or(Vebt_memberi2),X2),aTP_Lamm_nr(X2)),T),Ta,Xa2),(R3)),N3)
             => aa_nat_o(aa_o_fun_nat_o(aa_hea6792647905190977245_nat_o(aa_hea8977266003532396635_nat_o(aa_nat1261834975400290800_nat_o(aa_VEB7158420292131534498_nat_o(P,T),X2),Ta),Xa2),(R3)),N3) ) )
     => ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_Pro3916655985231737292Heap_o(produc770043135277712853Heap_o(vEBT_vebt_memberi),Xaa),H,H2),(R)),N)
       => aa_nat_o(aa_o_fun_nat_o(aa_hea6792647905190977245_nat_o(aa_hea8977266003532396635_nat_o(produc8468262472054731797_nat_o(P,Xaa),H),H2),(R)),N) ) ) ).

% vebt_memberi.raw_induct
tff(fact_6062_even__unset__bit__iff,axiom,
    ! [M: nat,A2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),bit_se4315839071623982667uint32(M,A2))
    <=> ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_6063_even__unset__bit__iff,axiom,
    ! [M: nat,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(M),A2))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_6064_even__unset__bit__iff,axiom,
    ! [M: nat,A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),bit_se5331074070815623765l_num1(M,A2))
    <=> ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_6065_even__unset__bit__iff,axiom,
    ! [M: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(M),A2))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_6066_even__unset__bit__iff,axiom,
    ! [M: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_6067_bezw__non__0,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y)
     => ( bezw(Xb,Y) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y))))) ) ) ).

% bezw_non_0
tff(fact_6068_unset__bit__Suc,axiom,
    ! [N: nat,A2: word_N3645301735248828278l_num1] : bit_se5331074070815623765l_num1(aa_nat_nat(suc,N),A2) = plus_p361126936061061375l_num1(modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),bit_se5331074070815623765l_num1(N,divide1791077408188789448l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6069_unset__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] : aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(aa_nat_nat(suc,N)),A2) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_se8260200283734997820nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6070_unset__bit__Suc,axiom,
    ! [N: nat,A2: uint32] : bit_se4315839071623982667uint32(aa_nat_nat(suc,N),A2) = plus_plus_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),bit_se4315839071623982667uint32(N,aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6071_unset__bit__Suc,axiom,
    ! [N: nat,A2: int] : aa_int_int(bit_se4203085406695923979it_int(aa_nat_nat(suc,N)),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6072_unset__bit__Suc,axiom,
    ! [N: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,N)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6073_tanh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_6074_VEBT__internal_Ovebt__succi_H_Omono,axiom,
    ! [Xb: produc3960855945107176009Ti_nat] : comple4513117127444775325on_nat(partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),heap_T7875578835903804844on_nat,aTP_Lamm_pi(Xb)) ).

% VEBT_internal.vebt_succi'.mono
tff(fact_6075_VEBT__internal_Ovebt__predi_H_Omono,axiom,
    ! [Xb: produc3960855945107176009Ti_nat] : comple4513117127444775325on_nat(partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),heap_T7875578835903804844on_nat,aTP_Lamm_pz(Xb)) ).

% VEBT_internal.vebt_predi'.mono
tff(fact_6076_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys),N) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_6077_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s7982070591426661849_VEBTi(Ys)))
     => ( nth_Pr316670251186196177_VEBTi(produc316462671093861988_VEBTi(Xs,Ys),N) = produc6084888613844515218_VEBTi(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s7982070591426661849_VEBTi(Ys))),nth_VEBT_VEBTi(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s7982070591426661849_VEBTi(Ys)))) ) ) ).

% product_nth
tff(fact_6078_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s7982070591426661849_VEBTi(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr8725177398587324397T_VEBT(produc1285381384045549624T_VEBT(Xs,Ys),N) = produc7053807326796202854T_VEBT(nth_VEBT_VEBTi(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_6079_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s7982070591426661849_VEBTi(Xs)),size_s7982070591426661849_VEBTi(Ys)))
     => ( nth_Pr6329974346453275474_VEBTi(produc194614972289024177_VEBTi(Xs,Ys),N) = produc436343169921013763_VEBTi(nth_VEBT_VEBTi(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s7982070591426661849_VEBTi(Ys))),nth_VEBT_VEBTi(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s7982070591426661849_VEBTi(Ys)))) ) ) ).

% product_nth
tff(fact_6080_product__nth,axiom,
    ! [N: nat,Xs: list_num,Ys: list_num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_num(Xs)),size_size_list_num(Ys)))
     => ( nth_Pr6456567536196504476um_num(product_num_num(Xs,Ys),N) = aa_num3156681643174609981um_num(product_Pair_num_num(aa_nat_num(nth_num(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_num(Ys)))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_num(Ys)))) ) ) ).

% product_nth
tff(fact_6081_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_real(Ys)))
     => ( nth_Pr6842391030413306568T_real(produc4908677263432625371T_real(Xs,Ys),N) = aa_rea4189226818901770745T_real(produc8117437818029410057T_real(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_real(Ys)))),aa_nat_real(nth_real(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_real(Ys)))) ) ) ).

% product_nth
tff(fact_6082_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Ys: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s7982070591426661849_VEBTi(Xs)),size_size_list_real(Ys)))
     => ( nth_Pr3433448822664029129i_real(produc5476717833281694120i_real(Xs,Ys),N) = produc8457151488442208762i_real(nth_VEBT_VEBTi(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_real(Ys))),aa_nat_real(nth_real(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_real(Ys)))) ) ) ).

% product_nth
tff(fact_6083_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr4606735188037164562VEBT_o(product_VEBT_VEBT_o(Xs,Ys),N) = aa_o_P2543913824914997935VEBT_o(produc8721562602347293563VEBT_o(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_o(Ys)))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_6084_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBTi,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s7982070591426661849_VEBTi(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr3306050735993963089EBTi_o(product_VEBT_VEBTi_o(Xs,Ys),N) = produc8194178580519725514EBTi_o(nth_VEBT_VEBTi(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_6085_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys),N) = aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_nat(Ys)))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_6086_Heap__Time__Monad_Obind__bind,axiom,
    ! [Fa: heap_Time_Heap_o,G: fun_o_2238389867833408620_VEBTi,K: fun_VE4765431424076933413_VEBTi] : heap_T1006145433769338483_VEBTi(heap_T5998771940306268294_VEBTi(Fa,G),K) = heap_T5998771940306268294_VEBTi(Fa,aa_fun432179723158315733_VEBTi(aTP_Lamm_qa(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6087_Heap__Time__Monad_Obind__bind,axiom,
    ! [Fa: heap_Time_Heap_nat,G: fun_na7848093160509028170_VEBTi,K: fun_VE4765431424076933413_VEBTi] : heap_T1006145433769338483_VEBTi(heap_T844888390831797134_VEBTi(Fa,G),K) = heap_T844888390831797134_VEBTi(Fa,aa_fun6311797650338102561_VEBTi(aTP_Lamm_qb(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6088_Heap__Time__Monad_Obind__bind,axiom,
    ! [Fa: heap_Time_Heap_nat,G: fun_na7848093160509028170_VEBTi,K: fun_VE7038733415900436564Heap_o] : heap_T3040810144269856602EBTi_o(heap_T844888390831797134_VEBTi(Fa,G),K) = heap_Time_bind_nat_o(Fa,aa_fun3725866701989745153Heap_o(aTP_Lamm_qc(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6089_Heap__Time__Monad_Obind__bind,axiom,
    ! [Fa: heap_Time_Heap_nat,G: fun_na2343073368905879023Heap_o,K: fun_o_8325599348076409421Heap_o] : heap_Time_bind_o_o(heap_Time_bind_nat_o(Fa,G),K) = heap_Time_bind_nat_o(Fa,aa_fun1108420279618462208Heap_o(aTP_Lamm_qd(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6090_Heap__Time__Monad_Obind__bind,axiom,
    ! [Fa: heap_Time_Heap_o,G: fun_o_8325599348076409421Heap_o,K: fun_o_2238389867833408620_VEBTi] : heap_T5998771940306268294_VEBTi(heap_Time_bind_o_o(Fa,G),K) = heap_T5998771940306268294_VEBTi(Fa,aa_fun5660287117450118806_VEBTi(aTP_Lamm_qe(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6091_Heap__Time__Monad_Obind__bind,axiom,
    ! [Fa: heap_Time_Heap_nat,G: fun_na2343073368905879023Heap_o,K: fun_o_2238389867833408620_VEBTi] : heap_T5998771940306268294_VEBTi(heap_Time_bind_nat_o(Fa,G),K) = heap_T844888390831797134_VEBTi(Fa,aa_fun5487288040762657696_VEBTi(aTP_Lamm_qf(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6092_Heap__Time__Monad_Obind__bind,axiom,
    ! [Fa: heap_Time_Heap_o,G: fun_o_4158591058828729991ap_nat,K: fun_na7848093160509028170_VEBTi] : heap_T844888390831797134_VEBTi(heap_Time_bind_o_nat(Fa,G),K) = heap_T5998771940306268294_VEBTi(Fa,aa_fun1800422248244663418_VEBTi(aTP_Lamm_qg(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6093_Heap__Time__Monad_Obind__bind,axiom,
    ! [Fa: heap_Time_Heap_nat,G: fun_na3533371890617757541ap_nat,K: fun_na7848093160509028170_VEBTi] : heap_T844888390831797134_VEBTi(heap_T7049098217575491753at_nat(Fa,G),K) = heap_T844888390831797134_VEBTi(Fa,aa_fun49764924992962236_VEBTi(aTP_Lamm_qh(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6094_Heap__Time__Monad_Obind__bind,axiom,
    ! [Fa: heap_Time_Heap_nat,G: fun_na3533371890617757541ap_nat,K: fun_na2343073368905879023Heap_o] : heap_Time_bind_nat_o(heap_T7049098217575491753at_nat(Fa,G),K) = heap_Time_bind_nat_o(Fa,aa_fun3424352875297450780Heap_o(aTP_Lamm_qi(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6095_Heap__Time__Monad_Obind__bind,axiom,
    ! [Fa: heap_T4980287057938770641_VEBTi,G: fun_li7499208429157053461_VEBTi,K: fun_VE4765431424076933413_VEBTi] : heap_T1006145433769338483_VEBTi(heap_T5877712393672139267_VEBTi(Fa,G),K) = heap_T5877712393672139267_VEBTi(Fa,aa_fun2525796888956276588_VEBTi(aTP_Lamm_qj(G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_6096_curry__conv,axiom,
    ! [Fa: fun_Pr5956696566359781750on_nat,A2: vEBT_VEBT,B2: vEBT_VEBTi] : aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(produc9121605208400662357on_nat(Fa),A2),B2) = aa_Pro1290670013494283871on_nat(Fa,produc6084888613844515218_VEBTi(A2,B2)) ).

% curry_conv
tff(fact_6097_curry__conv,axiom,
    ! [Fa: fun_Pr4877903221429077380Heap_o,A2: vEBT_VEBT,B2: vEBT_VEBTi] : aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(produc8906325494159015119Heap_o(Fa),A2),B2) = aa_Pro259339148018954309Heap_o(Fa,produc6084888613844515218_VEBTi(A2,B2)) ).

% curry_conv
tff(fact_6098_curry__conv,axiom,
    ! [Fa: fun_Pr4404027214751104635_VEBTi,A2: produc3625547720036274456_VEBTi,B2: nat] : aa_nat5408703056993926195_VEBTi(aa_Pro284253941917238388_VEBTi(produc2164094337957399884_VEBTi(Fa),A2),B2) = aa_Pro721367001817565044_VEBTi(Fa,produc1853644041309157249Ti_nat(A2,B2)) ).

% curry_conv
tff(fact_6099_curry__conv,axiom,
    ! [Fa: fun_Pr633315466666964198on_nat,A2: produc3625547720036274456_VEBTi,B2: nat] : aa_nat947733127081326238on_nat(aa_Pro1290670013494283871on_nat(produc1757988346207259447on_nat(Fa),A2),B2) = aa_Pro2727408364141080927on_nat(Fa,produc1853644041309157249Ti_nat(A2,B2)) ).

% curry_conv
tff(fact_6100_curry__conv,axiom,
    ! [Fa: fun_Pr940977554174145406Heap_o,A2: produc3625547720036274456_VEBTi,B2: nat] : aa_nat2910048432523430448Heap_o(aa_Pro259339148018954309Heap_o(produc8381543706267210711Heap_o(Fa),A2),B2) = aa_Pro6173831900156065711Heap_o(Fa,produc1853644041309157249Ti_nat(A2,B2)) ).

% curry_conv
tff(fact_6101_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_6102_curry__case__prod,axiom,
    ! [Fa: fun_VE3775134097089128360on_nat] : produc9121605208400662357on_nat(produc3521483158534984758on_nat(Fa)) = Fa ).

% curry_case_prod
tff(fact_6103_curry__case__prod,axiom,
    ! [Fa: fun_VE8181444127144381288Heap_o] : produc8906325494159015119Heap_o(produc844971584604743726Heap_o(Fa)) = Fa ).

% curry_case_prod
tff(fact_6104_curry__case__prod,axiom,
    ! [Fa: fun_Pr7184143894984516363_VEBTi] : produc2164094337957399884_VEBTi(produc2943724498215716011_VEBTi(Fa)) = Fa ).

% curry_case_prod
tff(fact_6105_curry__case__prod,axiom,
    ! [Fa: fun_Pr5956696566359781750on_nat] : produc1757988346207259447on_nat(produc183673358652719894on_nat(Fa)) = Fa ).

% curry_case_prod
tff(fact_6106_curry__case__prod,axiom,
    ! [Fa: fun_Pr4877903221429077380Heap_o] : produc8381543706267210711Heap_o(produc5872130906356439992Heap_o(Fa)) = Fa ).

% curry_case_prod
tff(fact_6107_curry__case__prod,axiom,
    ! [Fa: fun_na4138050610385705875at_nat] : produc6629854527392350932at_nat(produc2626176000494625587at_nat(Fa)) = Fa ).

% curry_case_prod
tff(fact_6108_curry__case__prod,axiom,
    ! [Fa: fun_nat_fun_nat_o] : produc1310100445399344235_nat_o(produc6081775807080527818_nat_o(Fa)) = Fa ).

% curry_case_prod
tff(fact_6109_curry__case__prod,axiom,
    ! [Fa: fun_in1696471567231710611nt_int] : produc8249235968001453780nt_int(produc4245557441103728435nt_int(Fa)) = Fa ).

% curry_case_prod
tff(fact_6110_curry__case__prod,axiom,
    ! [Fa: fun_int_fun_int_o] : produc175634133007206835_int_o(produc4947309494688390418_int_o(Fa)) = Fa ).

% curry_case_prod
tff(fact_6111_curry__case__prod,axiom,
    ! [Fa: fun_int_fun_int_int] : produc1016772743285680337nt_int(produc8211389475949308722nt_int(Fa)) = Fa ).

% curry_case_prod
tff(fact_6112_case__prod__curry,axiom,
    ! [Fa: fun_Pr5956696566359781750on_nat] : produc3521483158534984758on_nat(produc9121605208400662357on_nat(Fa)) = Fa ).

% case_prod_curry
tff(fact_6113_case__prod__curry,axiom,
    ! [Fa: fun_Pr4877903221429077380Heap_o] : produc844971584604743726Heap_o(produc8906325494159015119Heap_o(Fa)) = Fa ).

% case_prod_curry
tff(fact_6114_case__prod__curry,axiom,
    ! [Fa: fun_Pr4404027214751104635_VEBTi] : produc2943724498215716011_VEBTi(produc2164094337957399884_VEBTi(Fa)) = Fa ).

% case_prod_curry
tff(fact_6115_case__prod__curry,axiom,
    ! [Fa: fun_Pr633315466666964198on_nat] : produc183673358652719894on_nat(produc1757988346207259447on_nat(Fa)) = Fa ).

% case_prod_curry
tff(fact_6116_case__prod__curry,axiom,
    ! [Fa: fun_Pr940977554174145406Heap_o] : produc5872130906356439992Heap_o(produc8381543706267210711Heap_o(Fa)) = Fa ).

% case_prod_curry
tff(fact_6117_case__prod__curry,axiom,
    ! [Fa: fun_Pr2118491707810114563at_nat] : produc2626176000494625587at_nat(produc6629854527392350932at_nat(Fa)) = Fa ).

% case_prod_curry
tff(fact_6118_case__prod__curry,axiom,
    ! [Fa: fun_Pr4796943543331071514_nat_o] : produc6081775807080527818_nat_o(produc1310100445399344235_nat_o(Fa)) = Fa ).

% case_prod_curry
tff(fact_6119_case__prod__curry,axiom,
    ! [Fa: fun_Pr2478283820680522243nt_int] : produc4245557441103728435nt_int(produc8249235968001453780nt_int(Fa)) = Fa ).

% case_prod_curry
tff(fact_6120_case__prod__curry,axiom,
    ! [Fa: fun_Pr3238374608701176162_int_o] : produc4947309494688390418_int_o(produc175634133007206835_int_o(Fa)) = Fa ).

% case_prod_curry
tff(fact_6121_case__prod__curry,axiom,
    ! [Fa: fun_Pr982841511892225912nt_int] : produc8211389475949308722nt_int(produc1016772743285680337nt_int(Fa)) = Fa ).

% case_prod_curry
tff(fact_6122_tanh__real__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(tanh_real,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% tanh_real_zero_iff
tff(fact_6123_tanh__real__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xb)),aa_real_real(tanh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% tanh_real_less_iff
tff(fact_6124_tanh__real__neg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% tanh_real_neg_iff
tff(fact_6125_tanh__real__pos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tanh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% tanh_real_pos_iff
tff(fact_6126_tanh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% tanh_real_nonneg_iff
tff(fact_6127_tanh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% tanh_real_nonpos_iff
tff(fact_6128_length__product,axiom,
    ! [Xs: list_real,Ys: list_real] : size_s3932428310213730859l_real(product_real_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_real(Ys)) ).

% length_product
tff(fact_6129_length__product,axiom,
    ! [Xs: list_real,Ys: list_o] : size_s987546567493390085real_o(product_real_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_6130_length__product,axiom,
    ! [Xs: list_real,Ys: list_nat] : size_s1877336372972134351al_nat(product_real_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_6131_length__product,axiom,
    ! [Xs: list_real,Ys: list_int] : size_s8610625264895183403al_int(product_real_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_6132_length__product,axiom,
    ! [Xs: list_o,Ys: list_real] : size_s2624279037499656343o_real(product_o_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_real(Ys)) ).

% length_product
tff(fact_6133_length__product,axiom,
    ! [Xs: list_o,Ys: list_o] : size_s1515746228057227161od_o_o(product_o_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_6134_length__product,axiom,
    ! [Xs: list_o,Ys: list_nat] : size_s5443766701097040955_o_nat(product_o_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_6135_length__product,axiom,
    ! [Xs: list_o,Ys: list_int] : size_s2953683556165314199_o_int(product_o_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_6136_length__product,axiom,
    ! [Xs: list_nat,Ys: list_real] : size_s7910714270633306959t_real(product_nat_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_real(Ys)) ).

% length_product
tff(fact_6137_length__product,axiom,
    ! [Xs: list_nat,Ys: list_o] : size_s6491369823275344609_nat_o(product_nat_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_6138_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: fun_fu5786974620624140868ap_nat,C4: fun_na5845961453008041334_VEBTi] :
      ( comple3617025000840655848ap_nat(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T645984214031783516rd_nat,B5)
     => ( ! [Y2: nat] : comple7548324300139837517_VEBTi(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T7173139186834293313_VEBTi,aa_nat4209467000370889527_VEBTi(C4,Y2))
       => comple7548324300139837517_VEBTi(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T7173139186834293313_VEBTi,aa_fun90388479857626845_VEBTi(aTP_Lamm_ql(B5),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6139_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: fun_fu4424605311360796457_VEBTi,C4: fun_VE6491116402277797979_VEBTi] :
      ( comple7548324300139837517_VEBTi(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T7173139186834293313_VEBTi,B5)
     => ( ! [Y2: vEBT_VEBTi] : comple7548324300139837517_VEBTi(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T7173139186834293313_VEBTi,aa_VEB5816640871196325404_VEBTi(C4,Y2))
       => comple7548324300139837517_VEBTi(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T7173139186834293313_VEBTi,aa_fun4276395798348166338_VEBTi(aTP_Lamm_qn(B5),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6140_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: fun_fu5786974620624140868ap_nat,C4: fun_na8517577583218006873Heap_o] :
      ( comple3617025000840655848ap_nat(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T645984214031783516rd_nat,B5)
     => ( ! [Y2: nat] : comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aa_nat205861980368755650Heap_o(C4,Y2))
       => comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aa_fun3258944081180477171Heap_o(aTP_Lamm_qp(B5),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6141_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: fun_fu4424605311360796457_VEBTi,C4: fun_VE6943415934277719092Heap_o] :
      ( comple7548324300139837517_VEBTi(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T7173139186834293313_VEBTi,B5)
     => ( ! [Y2: vEBT_VEBTi] : comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aa_VEB6642844428796722333Heap_o(C4,Y2))
       => comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aa_fun5382224708498380120Heap_o(aTP_Lamm_qr(B5),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6142_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: fun_fu335202653675969872Heap_o,C4: fun_o_7171189317104958676_VEBTi] :
      ( comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,B5)
     => ( ! [Y2: $o] : comple7548324300139837517_VEBTi(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T7173139186834293313_VEBTi,aa_o_f8811364424157995653_VEBTi(C4,(Y2)))
       => comple7548324300139837517_VEBTi(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_T7173139186834293313_VEBTi,aa_fun2042045670669332929_VEBTi(aTP_Lamm_qt(B5),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6143_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: fun_fu335202653675969872Heap_o,C4: fun_o_2223814739247024763Heap_o] :
      ( comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,B5)
     => ( ! [Y2: $o] : comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aa_o_f6784619210364167924Heap_o(C4,(Y2)))
       => comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aa_fun4354595424708498391Heap_o(aTP_Lamm_qv(B5),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6144_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: fun_fu2606674006500944492ap_nat,C4: fun_na979242554834862325Heap_o] :
      ( comple6879160010309051114ap_nat(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T645984214031783516rd_nat,B5)
     => ( ! [Y2: nat] : comple3937103672976186900Heap_o(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_Time_Heap_ord_o,aa_nat5421218410817480758Heap_o(C4,Y2))
       => comple3937103672976186900Heap_o(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_Time_Heap_ord_o,aa_fun7768191950043460187Heap_o(aTP_Lamm_qx(B5),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6145_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: fun_fu2606674006500944492ap_nat,C4: fun_na7267038673063590618_VEBTi] :
      ( comple6879160010309051114ap_nat(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T645984214031783516rd_nat,B5)
     => ( ! [Y2: nat] : comple2136407771590434127_VEBTi(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T7173139186834293313_VEBTi,aa_nat5169694059731795907_VEBTi(C4,Y2))
       => comple2136407771590434127_VEBTi(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T7173139186834293313_VEBTi,aa_fun3907257510474286387_VEBTi(aTP_Lamm_qz(B5),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6146_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: fun_fu7983188961739674408Heap_o,C4: fun_o_5993975905899854163Heap_o] :
      ( comple3937103672976186900Heap_o(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_Time_Heap_ord_o,B5)
     => ( ! [Y2: $o] : comple3937103672976186900Heap_o(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_Time_Heap_ord_o,aa_o_f6675407086664040580Heap_o(C4,(Y2)))
       => comple3937103672976186900Heap_o(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_Time_Heap_ord_o,aa_fun8877585628723769663Heap_o(aTP_Lamm_rb(B5),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6147_Heap__Time__Monad_Obind__mono,axiom,
    ! [B5: fun_fu7983188961739674408Heap_o,C4: fun_o_2166024663187074044_VEBTi] :
      ( comple3937103672976186900Heap_o(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_Time_Heap_ord_o,B5)
     => ( ! [Y2: $o] : comple2136407771590434127_VEBTi(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T7173139186834293313_VEBTi,aa_o_f1510940835853032693_VEBTi(C4,(Y2)))
       => comple2136407771590434127_VEBTi(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T7173139186834293313_VEBTi,aa_fun565820801828592151_VEBTi(aTP_Lamm_rd(B5),C4)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_6148_heap_Oconst__mono,axiom,
    ! [Ord: fun_fu7962274784768186808_nat_o,C: heap_T2636463487746394924on_nat] : comple7427593846244870586on_nat(Ord,heap_T7875578835903804844on_nat,aTP_Lamm_re(C)) ).

% heap.const_mono
tff(fact_6149_heap_Oconst__mono,axiom,
    ! [Ord: fun_fu2183070276626753582ap_o_o,C: heap_Time_Heap_o] : comple6933591415923782038Heap_o(Ord,heap_Time_Heap_ord_o,aTP_Lamm_rf(C)) ).

% heap.const_mono
tff(fact_6150_heap_Oconst__mono,axiom,
    ! [Ord: fun_fu3145514261243071828EBTi_o,C: heap_T8145700208782473153_VEBTi] : comple422786390015765981_VEBTi(Ord,heap_T7173139186834293313_VEBTi,aTP_Lamm_rg(C)) ).

% heap.const_mono
tff(fact_6151_heap_Oconst__mono,axiom,
    ! [Ord: fun_fu5284765496190745342_nat_o,C: heap_T2636463487746394924on_nat] : comple4513117127444775325on_nat(Ord,heap_T7875578835903804844on_nat,aTP_Lamm_rh(C)) ).

% heap.const_mono
tff(fact_6152_heap_Oconst__mono,axiom,
    ! [Ord: fun_fu6850233865993928942ap_o_o,C: heap_Time_Heap_o] : comple5241396456474302457Heap_o(Ord,heap_Time_Heap_ord_o,aTP_Lamm_ri(C)) ).

% heap.const_mono
tff(fact_6153_curry__K,axiom,
    ! [C: fun_na5556145776640407093on_nat,X3: vEBT_VEBT,Xa: vEBT_VEBTi] : aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(produc9121605208400662357on_nat(aTP_Lamm_rj(C)),X3),Xa) = C ).

% curry_K
tff(fact_6154_curry__K,axiom,
    ! [C: fun_na2343073368905879023Heap_o,X3: vEBT_VEBT,Xa: vEBT_VEBTi] : aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(produc8906325494159015119Heap_o(aTP_Lamm_rk(C)),X3),Xa) = C ).

% curry_K
tff(fact_6155_curry__K,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,X3: produc3625547720036274456_VEBTi,Xa: nat] : aa_nat5408703056993926195_VEBTi(aa_Pro284253941917238388_VEBTi(produc2164094337957399884_VEBTi(aTP_Lamm_rl(C)),X3),Xa) = C ).

% curry_K
tff(fact_6156_curry__K,axiom,
    ! [C: heap_T2636463487746394924on_nat,X3: produc3625547720036274456_VEBTi,Xa: nat] : aa_nat947733127081326238on_nat(aa_Pro1290670013494283871on_nat(produc1757988346207259447on_nat(aTP_Lamm_rm(C)),X3),Xa) = C ).

% curry_K
tff(fact_6157_curry__K,axiom,
    ! [C: heap_Time_Heap_o,X3: produc3625547720036274456_VEBTi,Xa: nat] : aa_nat2910048432523430448Heap_o(aa_Pro259339148018954309Heap_o(produc8381543706267210711Heap_o(aTP_Lamm_rn(C)),X3),Xa) = C ).

% curry_K
tff(fact_6158_tanh__real__lt__1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xb)),one_one_real) ).

% tanh_real_lt_1
tff(fact_6159_curry__def,axiom,
    ! [X3: fun_Pr5956696566359781750on_nat,Xa: vEBT_VEBT,Xb3: vEBT_VEBTi] : aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(produc9121605208400662357on_nat(X3),Xa),Xb3) = aa_Pro1290670013494283871on_nat(X3,produc6084888613844515218_VEBTi(Xa,Xb3)) ).

% curry_def
tff(fact_6160_curry__def,axiom,
    ! [X3: fun_Pr4877903221429077380Heap_o,Xa: vEBT_VEBT,Xb3: vEBT_VEBTi] : aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(produc8906325494159015119Heap_o(X3),Xa),Xb3) = aa_Pro259339148018954309Heap_o(X3,produc6084888613844515218_VEBTi(Xa,Xb3)) ).

% curry_def
tff(fact_6161_curry__def,axiom,
    ! [X3: fun_Pr4404027214751104635_VEBTi,Xa: produc3625547720036274456_VEBTi,Xb3: nat] : aa_nat5408703056993926195_VEBTi(aa_Pro284253941917238388_VEBTi(produc2164094337957399884_VEBTi(X3),Xa),Xb3) = aa_Pro721367001817565044_VEBTi(X3,produc1853644041309157249Ti_nat(Xa,Xb3)) ).

% curry_def
tff(fact_6162_curry__def,axiom,
    ! [X3: fun_Pr633315466666964198on_nat,Xa: produc3625547720036274456_VEBTi,Xb3: nat] : aa_nat947733127081326238on_nat(aa_Pro1290670013494283871on_nat(produc1757988346207259447on_nat(X3),Xa),Xb3) = aa_Pro2727408364141080927on_nat(X3,produc1853644041309157249Ti_nat(Xa,Xb3)) ).

% curry_def
tff(fact_6163_curry__def,axiom,
    ! [X3: fun_Pr940977554174145406Heap_o,Xa: produc3625547720036274456_VEBTi,Xb3: nat] : aa_nat2910048432523430448Heap_o(aa_Pro259339148018954309Heap_o(produc8381543706267210711Heap_o(X3),Xa),Xb3) = aa_Pro6173831900156065711Heap_o(X3,produc1853644041309157249Ti_nat(Xa,Xb3)) ).

% curry_def
tff(fact_6164_fold__if__return,axiom,
    ! [B2: $o,C: option_nat,D: option_nat] :
      $ite((B2),heap_T3487192422709364219on_nat(C),heap_T3487192422709364219on_nat(D)) = heap_T3487192422709364219on_nat($ite((B2),C,D)) ).

% fold_if_return
tff(fact_6165_fold__if__return,axiom,
    ! [B2: $o,C: nat,D: nat] :
      $ite((B2),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,C),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,D)) = aa_nat5821310369566504398ap_nat(heap_Time_return_nat,
        $ite((B2),C,D)) ).

% fold_if_return
tff(fact_6166_fold__if__return,axiom,
    ! [B2: $o,C: $o,D: $o] :
      $ite((B2),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,(C)),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,(D))) = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,
        $ite((B2),(C),(D))) ).

% fold_if_return
tff(fact_6167_fold__if__return,axiom,
    ! [B2: $o,C: vEBT_VEBTi,D: vEBT_VEBTi] :
      $ite((B2),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,C),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,D)) = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,
        $ite((B2),C,D)) ).

% fold_if_return
tff(fact_6168_effect__deterministic_I3_J,axiom,
    ! [Fa: heap_T2636463487746394924on_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,A2: option_nat,N: nat,H7: heap_e7401611519738050253t_unit,B2: option_nat,N5: nat] :
      ( aa_nat_o(heap_T306965388786959644on_nat(Fa,H,H2,A2),N)
     => ( aa_nat_o(heap_T306965388786959644on_nat(Fa,H,H7,B2),N5)
       => ( N = N5 ) ) ) ).

% effect_deterministic(3)
tff(fact_6169_effect__deterministic_I3_J,axiom,
    ! [Fa: heap_Time_Heap_o,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,A2: $o,N: nat,H7: heap_e7401611519738050253t_unit,B2: $o,N5: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(Fa,H,H2),(A2)),N)
     => ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(Fa,H,H7),(B2)),N5)
       => ( N = N5 ) ) ) ).

% effect_deterministic(3)
tff(fact_6170_effect__deterministic_I3_J,axiom,
    ! [Fa: heap_T8145700208782473153_VEBTi,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,A2: vEBT_VEBTi,N: nat,H7: heap_e7401611519738050253t_unit,B2: vEBT_VEBTi,N5: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(Fa,H,H2),A2),N)
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(Fa,H,H7),B2),N5)
       => ( N = N5 ) ) ) ).

% effect_deterministic(3)
tff(fact_6171_effect__deterministic_I2_J,axiom,
    ! [Fa: heap_T2636463487746394924on_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,A2: option_nat,N: nat,H7: heap_e7401611519738050253t_unit,B2: option_nat,N5: nat] :
      ( aa_nat_o(heap_T306965388786959644on_nat(Fa,H,H2,A2),N)
     => ( aa_nat_o(heap_T306965388786959644on_nat(Fa,H,H7,B2),N5)
       => ( H2 = H7 ) ) ) ).

% effect_deterministic(2)
tff(fact_6172_effect__deterministic_I2_J,axiom,
    ! [Fa: heap_Time_Heap_o,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,A2: $o,N: nat,H7: heap_e7401611519738050253t_unit,B2: $o,N5: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(Fa,H,H2),(A2)),N)
     => ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(Fa,H,H7),(B2)),N5)
       => ( H2 = H7 ) ) ) ).

% effect_deterministic(2)
tff(fact_6173_effect__deterministic_I2_J,axiom,
    ! [Fa: heap_T8145700208782473153_VEBTi,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,A2: vEBT_VEBTi,N: nat,H7: heap_e7401611519738050253t_unit,B2: vEBT_VEBTi,N5: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(Fa,H,H2),A2),N)
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(Fa,H,H7),B2),N5)
       => ( H2 = H7 ) ) ) ).

% effect_deterministic(2)
tff(fact_6174_effect__deterministic_I1_J,axiom,
    ! [Fa: heap_T2636463487746394924on_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,A2: option_nat,N: nat,H7: heap_e7401611519738050253t_unit,B2: option_nat,N5: nat] :
      ( aa_nat_o(heap_T306965388786959644on_nat(Fa,H,H2,A2),N)
     => ( aa_nat_o(heap_T306965388786959644on_nat(Fa,H,H7,B2),N5)
       => ( A2 = B2 ) ) ) ).

% effect_deterministic(1)
tff(fact_6175_effect__deterministic_I1_J,axiom,
    ! [Fa: heap_Time_Heap_o,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,A2: $o,N: nat,H7: heap_e7401611519738050253t_unit,B2: $o,N5: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(Fa,H,H2),(A2)),N)
     => ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(Fa,H,H7),(B2)),N5)
       => ( (A2)
        <=> (B2) ) ) ) ).

% effect_deterministic(1)
tff(fact_6176_effect__deterministic_I1_J,axiom,
    ! [Fa: heap_T8145700208782473153_VEBTi,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,A2: vEBT_VEBTi,N: nat,H7: heap_e7401611519738050253t_unit,B2: vEBT_VEBTi,N5: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(Fa,H,H2),A2),N)
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(Fa,H,H7),B2),N5)
       => ( A2 = B2 ) ) ) ).

% effect_deterministic(1)
tff(fact_6177_effect__ifE,axiom,
    ! [C: $o,Tb: heap_T2636463487746394924on_nat,E2: heap_T2636463487746394924on_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: option_nat,N: nat] :
      ( aa_nat_o(heap_T306965388786959644on_nat($ite((C),Tb,E2),H,H2,R),N)
     => ( ( (C)
         => ~ aa_nat_o(heap_T306965388786959644on_nat(Tb,H,H2,R),N) )
       => ~ ( ~ (C)
           => ~ aa_nat_o(heap_T306965388786959644on_nat(E2,H,H2,R),N) ) ) ) ).

% effect_ifE
tff(fact_6178_effect__ifE,axiom,
    ! [C: $o,Tb: heap_Time_Heap_o,E2: heap_Time_Heap_o,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: $o,N: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o($ite((C),Tb,E2),H,H2),(R)),N)
     => ( ( (C)
         => ~ aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(Tb,H,H2),(R)),N) )
       => ~ ( ~ (C)
           => ~ aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(E2,H,H2),(R)),N) ) ) ) ).

% effect_ifE
tff(fact_6179_effect__ifE,axiom,
    ! [C: $o,Tb: heap_T8145700208782473153_VEBTi,E2: heap_T8145700208782473153_VEBTi,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: vEBT_VEBTi,N: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi($ite((C),Tb,E2),H,H2),R),N)
     => ( ( (C)
         => ~ aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(Tb,H,H2),R),N) )
       => ~ ( ~ (C)
           => ~ aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(E2,H,H2),R),N) ) ) ) ).

% effect_ifE
tff(fact_6180_effect__ifI,axiom,
    ! [C: $o,Tb: heap_T2636463487746394924on_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: option_nat,N: nat,E2: heap_T2636463487746394924on_nat] :
      ( ( (C)
       => aa_nat_o(heap_T306965388786959644on_nat(Tb,H,H2,R),N) )
     => ( ( ~ (C)
         => aa_nat_o(heap_T306965388786959644on_nat(E2,H,H2,R),N) )
       => aa_nat_o(heap_T306965388786959644on_nat($ite((C),Tb,E2),H,H2,R),N) ) ) ).

% effect_ifI
tff(fact_6181_effect__ifI,axiom,
    ! [C: $o,Tb: heap_Time_Heap_o,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: $o,N: nat,E2: heap_Time_Heap_o] :
      ( ( (C)
       => aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(Tb,H,H2),(R)),N) )
     => ( ( ~ (C)
         => aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(E2,H,H2),(R)),N) )
       => aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o($ite((C),Tb,E2),H,H2),(R)),N) ) ) ).

% effect_ifI
tff(fact_6182_effect__ifI,axiom,
    ! [C: $o,Tb: heap_T8145700208782473153_VEBTi,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: vEBT_VEBTi,N: nat,E2: heap_T8145700208782473153_VEBTi] :
      ( ( (C)
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(Tb,H,H2),R),N) )
     => ( ( ~ (C)
         => aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(E2,H,H2),R),N) )
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi($ite((C),Tb,E2),H,H2),R),N) ) ) ).

% effect_ifI
tff(fact_6183_distrib__if__bind,axiom,
    ! [B2: $o,C: heap_T4980287057938770641_VEBTi,D: heap_T4980287057938770641_VEBTi,Fa: fun_li7499208429157053461_VEBTi] :
      heap_T5877712393672139267_VEBTi($ite((B2),C,D),Fa) = $ite((B2),heap_T5877712393672139267_VEBTi(C,Fa),heap_T5877712393672139267_VEBTi(D,Fa)) ).

% distrib_if_bind
tff(fact_6184_distrib__if__bind,axiom,
    ! [B2: $o,C: heap_Time_Heap_o,D: heap_Time_Heap_o,Fa: fun_o_2238389867833408620_VEBTi] :
      heap_T5998771940306268294_VEBTi($ite((B2),C,D),Fa) = $ite((B2),heap_T5998771940306268294_VEBTi(C,Fa),heap_T5998771940306268294_VEBTi(D,Fa)) ).

% distrib_if_bind
tff(fact_6185_distrib__if__bind,axiom,
    ! [B2: $o,C: heap_Time_Heap_nat,D: heap_Time_Heap_nat,Fa: fun_na7848093160509028170_VEBTi] :
      heap_T844888390831797134_VEBTi($ite((B2),C,D),Fa) = $ite((B2),heap_T844888390831797134_VEBTi(C,Fa),heap_T844888390831797134_VEBTi(D,Fa)) ).

% distrib_if_bind
tff(fact_6186_distrib__if__bind,axiom,
    ! [B2: $o,C: heap_Time_Heap_nat,D: heap_Time_Heap_nat,Fa: fun_na5556145776640407093on_nat] :
      heap_T8222160169144143993on_nat($ite((B2),C,D),Fa) = $ite((B2),heap_T8222160169144143993on_nat(C,Fa),heap_T8222160169144143993on_nat(D,Fa)) ).

% distrib_if_bind
tff(fact_6187_distrib__if__bind,axiom,
    ! [B2: $o,C: heap_Time_Heap_nat,D: heap_Time_Heap_nat,Fa: fun_na2343073368905879023Heap_o] :
      heap_Time_bind_nat_o($ite((B2),C,D),Fa) = $ite((B2),heap_Time_bind_nat_o(C,Fa),heap_Time_bind_nat_o(D,Fa)) ).

% distrib_if_bind
tff(fact_6188_effect__returnI,axiom,
    ! [H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,Xb: option_nat] :
      ( ( H = H2 )
     => aa_nat_o(heap_T306965388786959644on_nat(heap_T3487192422709364219on_nat(Xb),H,H2,Xb),one_one_nat) ) ).

% effect_returnI
tff(fact_6189_effect__returnI,axiom,
    ! [H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,Xb: nat] :
      ( ( H = H2 )
     => aa_nat_o(aa_nat_fun_nat_o(heap_Time_effect_nat(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Xb),H,H2),Xb),one_one_nat) ) ).

% effect_returnI
tff(fact_6190_effect__returnI,axiom,
    ! [H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,Xb: $o] :
      ( ( H = H2 )
     => aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,(Xb)),H,H2),(Xb)),one_one_nat) ) ).

% effect_returnI
tff(fact_6191_effect__returnI,axiom,
    ! [H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,Xb: vEBT_VEBTi] :
      ( ( H = H2 )
     => aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Xb),H,H2),Xb),one_one_nat) ) ).

% effect_returnI
tff(fact_6192_effect__returnE,axiom,
    ! [Xb: option_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: option_nat,N: nat] :
      ( aa_nat_o(heap_T306965388786959644on_nat(heap_T3487192422709364219on_nat(Xb),H,H2,R),N)
     => ~ ( ( R = Xb )
         => ( ( H2 = H )
           => ( N != one_one_nat ) ) ) ) ).

% effect_returnE
tff(fact_6193_effect__returnE,axiom,
    ! [Xb: nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(heap_Time_effect_nat(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Xb),H,H2),R),N)
     => ~ ( ( R = Xb )
         => ( ( H2 = H )
           => ( N != one_one_nat ) ) ) ) ).

% effect_returnE
tff(fact_6194_effect__returnE,axiom,
    ! [Xb: $o,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: $o,N: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,(Xb)),H,H2),(R)),N)
     => ~ ( ( (R)
          <=> (Xb) )
         => ( ( H2 = H )
           => ( N != one_one_nat ) ) ) ) ).

% effect_returnE
tff(fact_6195_effect__returnE,axiom,
    ! [Xb: vEBT_VEBTi,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: vEBT_VEBTi,N: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Xb),H,H2),R),N)
     => ~ ( ( R = Xb )
         => ( ( H2 = H )
           => ( N != one_one_nat ) ) ) ) ).

% effect_returnE
tff(fact_6196_effect__bindE,axiom,
    ! [Fa: heap_Time_Heap_nat,G: fun_na2343073368905879023Heap_o,H: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: $o,N: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(heap_Time_bind_nat_o(Fa,G),H,H7),(R5)),N)
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: nat,N1: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(heap_Time_effect_nat(Fa,H,H8),R3),N1)
           => ! [N22: nat] :
                ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_nat2910048432523430448Heap_o(G,R3),H8,H7),(R5)),N22)
               => ( N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6197_effect__bindE,axiom,
    ! [Fa: heap_Time_Heap_o,G: fun_o_8325599348076409421Heap_o,H: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: $o,N: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(heap_Time_bind_o_o(Fa,G),H,H7),(R5)),N)
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: $o,N1: nat] :
            ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(Fa,H,H8),(R3)),N1)
           => ! [N22: nat] :
                ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_o_H7399033255965945470Heap_o(G,(R3)),H8,H7),(R5)),N22)
               => ( N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6198_effect__bindE,axiom,
    ! [Fa: heap_T8145700208782473153_VEBTi,G: fun_VE7038733415900436564Heap_o,H: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: $o,N: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(heap_T3040810144269856602EBTi_o(Fa,G),H,H7),(R5)),N)
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N1: nat] :
            ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(Fa,H,H8),R3),N1)
           => ! [N22: nat] :
                ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_VEB8194921341634505877Heap_o(G,R3),H8,H7),(R5)),N22)
               => ( N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6199_effect__bindE,axiom,
    ! [Fa: heap_Time_Heap_nat,G: fun_na7848093160509028170_VEBTi,H: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T844888390831797134_VEBTi(Fa,G),H,H7),R5),N)
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: nat,N1: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(heap_Time_effect_nat(Fa,H,H8),R3),N1)
           => ! [N22: nat] :
                ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_nat5408703056993926195_VEBTi(G,R3),H8,H7),R5),N22)
               => ( N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6200_effect__bindE,axiom,
    ! [Fa: heap_Time_Heap_o,G: fun_o_2238389867833408620_VEBTi,H: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T5998771940306268294_VEBTi(Fa,G),H,H7),R5),N)
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: $o,N1: nat] :
            ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(Fa,H,H8),(R3)),N1)
           => ! [N22: nat] :
                ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_o_H7577119287286566245_VEBTi(G,(R3)),H8,H7),R5),N22)
               => ( N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6201_effect__bindE,axiom,
    ! [Fa: heap_T8145700208782473153_VEBTi,G: fun_VE4765431424076933413_VEBTi,H: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T1006145433769338483_VEBTi(Fa,G),H,H7),R5),N)
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N1: nat] :
            ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(Fa,H,H8),R3),N1)
           => ! [N22: nat] :
                ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_VEB836174949890266638_VEBTi(G,R3),H8,H7),R5),N22)
               => ( N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6202_effect__bindE,axiom,
    ! [Fa: heap_Time_Heap_nat,G: fun_na5556145776640407093on_nat,H: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: option_nat,N: nat] :
      ( aa_nat_o(heap_T306965388786959644on_nat(heap_T8222160169144143993on_nat(Fa,G),H,H7,R5),N)
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: nat,N1: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(heap_Time_effect_nat(Fa,H,H8),R3),N1)
           => ! [N22: nat] :
                ( aa_nat_o(heap_T306965388786959644on_nat(aa_nat947733127081326238on_nat(G,R3),H8,H7,R5),N22)
               => ( N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6203_effect__bindE,axiom,
    ! [Fa: heap_Time_Heap_o,G: fun_o_1929998731553720919on_nat,H: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: option_nat,N: nat] :
      ( aa_nat_o(heap_T306965388786959644on_nat(heap_T6306279297776390513on_nat(Fa,G),H,H7,R5),N)
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: $o,N1: nat] :
            ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(Fa,H,H8),(R3)),N1)
           => ! [N22: nat] :
                ( aa_nat_o(heap_T306965388786959644on_nat(aa_o_H4215342640551218896on_nat(G,(R3)),H8,H7,R5),N22)
               => ( N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6204_effect__bindE,axiom,
    ! [Fa: heap_T8145700208782473153_VEBTi,G: fun_VE2693554162914142864on_nat,H: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: option_nat,N: nat] :
      ( aa_nat_o(heap_T306965388786959644on_nat(heap_T2868974464944644318on_nat(Fa,G),H,H7,R5),N)
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N1: nat] :
            ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(Fa,H,H8),R3),N1)
           => ! [N22: nat] :
                ( aa_nat_o(heap_T306965388786959644on_nat(aa_VEB2711320435582112505on_nat(G,R3),H8,H7,R5),N22)
               => ( N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6205_effect__bindE,axiom,
    ! [Fa: heap_T2636463487746394924on_nat,G: fun_op3694375823608013759Heap_o,H: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: $o,N: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(heap_T6471384023045698863_nat_o(Fa,G),H,H7),(R5)),N)
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: option_nat,N1: nat] :
            ( aa_nat_o(heap_T306965388786959644on_nat(Fa,H,H8,R3),N1)
           => ! [N22: nat] :
                ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_opt6753654531493786624Heap_o(G,R3),H8,H7),(R5)),N22)
               => ( N != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_6206_effect__bindI,axiom,
    ! [Fa: heap_Time_Heap_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: nat,N: nat,G: fun_na2343073368905879023Heap_o,H7: heap_e7401611519738050253t_unit,R5: $o,N5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(heap_Time_effect_nat(Fa,H,H2),R),N)
     => ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_nat2910048432523430448Heap_o(G,R),H2,H7),(R5)),N5)
       => aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(heap_Time_bind_nat_o(Fa,G),H,H7),(R5)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N5)) ) ) ).

% effect_bindI
tff(fact_6207_effect__bindI,axiom,
    ! [Fa: heap_Time_Heap_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: nat,N: nat,G: fun_na7848093160509028170_VEBTi,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(heap_Time_effect_nat(Fa,H,H2),R),N)
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_nat5408703056993926195_VEBTi(G,R),H2,H7),R5),N5)
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T844888390831797134_VEBTi(Fa,G),H,H7),R5),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N5)) ) ) ).

% effect_bindI
tff(fact_6208_effect__bindI,axiom,
    ! [Fa: heap_Time_Heap_o,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: $o,N: nat,G: fun_o_8325599348076409421Heap_o,H7: heap_e7401611519738050253t_unit,R5: $o,N5: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(Fa,H,H2),(R)),N)
     => ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_o_H7399033255965945470Heap_o(G,(R)),H2,H7),(R5)),N5)
       => aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(heap_Time_bind_o_o(Fa,G),H,H7),(R5)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N5)) ) ) ).

% effect_bindI
tff(fact_6209_effect__bindI,axiom,
    ! [Fa: heap_Time_Heap_o,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: $o,N: nat,G: fun_o_2238389867833408620_VEBTi,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N5: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(Fa,H,H2),(R)),N)
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_o_H7577119287286566245_VEBTi(G,(R)),H2,H7),R5),N5)
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T5998771940306268294_VEBTi(Fa,G),H,H7),R5),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N5)) ) ) ).

% effect_bindI
tff(fact_6210_effect__bindI,axiom,
    ! [Fa: heap_T8145700208782473153_VEBTi,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: vEBT_VEBTi,N: nat,G: fun_VE7038733415900436564Heap_o,H7: heap_e7401611519738050253t_unit,R5: $o,N5: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(Fa,H,H2),R),N)
     => ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_VEB8194921341634505877Heap_o(G,R),H2,H7),(R5)),N5)
       => aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(heap_T3040810144269856602EBTi_o(Fa,G),H,H7),(R5)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N5)) ) ) ).

% effect_bindI
tff(fact_6211_effect__bindI,axiom,
    ! [Fa: heap_T8145700208782473153_VEBTi,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: vEBT_VEBTi,N: nat,G: fun_VE4765431424076933413_VEBTi,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N5: nat] :
      ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(Fa,H,H2),R),N)
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_VEB836174949890266638_VEBTi(G,R),H2,H7),R5),N5)
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T1006145433769338483_VEBTi(Fa,G),H,H7),R5),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N5)) ) ) ).

% effect_bindI
tff(fact_6212_effect__bindI,axiom,
    ! [Fa: heap_Time_Heap_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: nat,N: nat,G: fun_na5556145776640407093on_nat,H7: heap_e7401611519738050253t_unit,R5: option_nat,N5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(heap_Time_effect_nat(Fa,H,H2),R),N)
     => ( aa_nat_o(heap_T306965388786959644on_nat(aa_nat947733127081326238on_nat(G,R),H2,H7,R5),N5)
       => aa_nat_o(heap_T306965388786959644on_nat(heap_T8222160169144143993on_nat(Fa,G),H,H7,R5),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N5)) ) ) ).

% effect_bindI
tff(fact_6213_effect__bindI,axiom,
    ! [Fa: heap_T4980287057938770641_VEBTi,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: list_VEBT_VEBTi,N: nat,G: fun_li7499208429157053461_VEBTi,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N5: nat] :
      ( aa_nat_o(heap_T33481931004607297_VEBTi(Fa,H,H2,R),N)
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_lis1839255523752068350_VEBTi(G,R),H2,H7),R5),N5)
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T5877712393672139267_VEBTi(Fa,G),H,H7),R5),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N5)) ) ) ).

% effect_bindI
tff(fact_6214_effect__bindI,axiom,
    ! [Fa: heap_T2636463487746394924on_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: option_nat,N: nat,G: fun_op3694375823608013759Heap_o,H7: heap_e7401611519738050253t_unit,R5: $o,N5: nat] :
      ( aa_nat_o(heap_T306965388786959644on_nat(Fa,H,H2,R),N)
     => ( aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(aa_opt6753654531493786624Heap_o(G,R),H2,H7),(R5)),N5)
       => aa_nat_o(aa_o_fun_nat_o(heap_Time_effect_o(heap_T6471384023045698863_nat_o(Fa,G),H,H7),(R5)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N5)) ) ) ).

% effect_bindI
tff(fact_6215_effect__bindI,axiom,
    ! [Fa: heap_T2636463487746394924on_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: option_nat,N: nat,G: fun_op1509349959255590266_VEBTi,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N5: nat] :
      ( aa_nat_o(heap_T306965388786959644on_nat(Fa,H,H2,R),N)
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_opt6483592524530923107_VEBTi(G,R),H2,H7),R5),N5)
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(heap_T5661892481228163294_VEBTi(Fa,G),H,H7),R5),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N5)) ) ) ).

% effect_bindI
tff(fact_6216_VEBT__internal_Ovebt__predi_H_Ofixp__induct,axiom,
    ! [P: fun_fu6711510597495717193_nat_o] :
      ( comple5664458124825824345on_nat(partia6039416512482706817Ti_nat(heap_T7048022066654196708on_nat),partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),aTP_Lamm_ro(P))
     => ( aa_fun5369152942363543986_nat_o(P,aTP_Lamm_rq)
       => ( ! [F3: fun_VE3775134097089128360on_nat] :
              ( aa_fun5369152942363543986_nat_o(P,F3)
             => aa_fun5369152942363543986_nat_o(P,aTP_Lamm_rr(F3)) )
         => aa_fun5369152942363543986_nat_o(P,vEBT_VEBT_vebt_predi) ) ) ) ).

% VEBT_internal.vebt_predi'.fixp_induct
tff(fact_6217_VEBT__internal_Ovebt__succi_H_Ofixp__induct,axiom,
    ! [P: fun_fu6711510597495717193_nat_o] :
      ( comple5664458124825824345on_nat(partia6039416512482706817Ti_nat(heap_T7048022066654196708on_nat),partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),aTP_Lamm_ro(P))
     => ( aa_fun5369152942363543986_nat_o(P,aTP_Lamm_rq)
       => ( ! [F3: fun_VE3775134097089128360on_nat] :
              ( aa_fun5369152942363543986_nat_o(P,F3)
             => aa_fun5369152942363543986_nat_o(P,aTP_Lamm_rs(F3)) )
         => aa_fun5369152942363543986_nat_o(P,vEBT_VEBT_vebt_succi) ) ) ) ).

% VEBT_internal.vebt_succi'.fixp_induct
tff(fact_6218_vebt__succi_Omono,axiom,
    ! [Xb: produc3881548065746020326Ti_nat] : comple7427593846244870586on_nat(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T7875578835903804844on_nat,aTP_Lamm_sa(Xb)) ).

% vebt_succi.mono
tff(fact_6219_vebt__predi_Omono,axiom,
    ! [Xb: produc3881548065746020326Ti_nat] : comple7427593846244870586on_nat(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T7875578835903804844on_nat,aTP_Lamm_si(Xb)) ).

% vebt_predi.mono
tff(fact_6220_pred__subset__eq2,axiom,
    ! [R4: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ord_le3306795289226699318_num_o(aTP_Lamm_sj(R4),aTP_Lamm_sj(S3))
    <=> ord_le880128212290418581um_num(R4,S3) ) ).

% pred_subset_eq2
tff(fact_6221_pred__subset__eq2,axiom,
    ! [R4: set_Pr563407847431865468T_VEBT,S3: set_Pr563407847431865468T_VEBT] :
      ( ord_le2619216865166325287VEBT_o(aTP_Lamm_sk(R4),aTP_Lamm_sk(S3))
    <=> ord_le6438908469242860764T_VEBT(R4,S3) ) ).

% pred_subset_eq2
tff(fact_6222_pred__subset__eq2,axiom,
    ! [R4: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ord_le1526520017675560704_num_o(aTP_Lamm_sl(R4),aTP_Lamm_sl(S3))
    <=> ord_le8085105155179020875at_num(R4,S3) ) ).

% pred_subset_eq2
tff(fact_6223_pred__subset__eq2,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ord_le768339454706049738_nat_o(aTP_Lamm_sm(R4),aTP_Lamm_sm(S3))
    <=> ord_le3146513528884898305at_nat(R4,S3) ) ).

% pred_subset_eq2
tff(fact_6224_pred__subset__eq2,axiom,
    ! [R4: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ord_le77980682512674322_int_o(aTP_Lamm_sn(R4),aTP_Lamm_sn(S3))
    <=> ord_le2843351958646193337nt_int(R4,S3) ) ).

% pred_subset_eq2
tff(fact_6225_ln__one__minus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,Xb)),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(minus_minus_real(one_one_real),Xb))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_6226_curryI,axiom,
    ! [Fa: fun_Pr6331313853612757382_num_o,A2: num,B2: num] :
      ( aa_Pro7685007914727650815_num_o(Fa,aa_num3156681643174609981um_num(product_Pair_num_num(A2),B2))
     => aa_num_o(aa_num_fun_num_o(produc932273227547478743_num_o(Fa),A2),B2) ) ).

% curryI
tff(fact_6227_curryI,axiom,
    ! [Fa: fun_Pr6320724876807202603VEBT_o,A2: nat,B2: produc4813437837504472865T_VEBT] :
      ( aa_Pro1497393596385541012VEBT_o(Fa,aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A2),B2))
     => aa_Pro3461009536670225267VEBT_o(aa_nat6863640065291604488VEBT_o(produc1960320730580999890VEBT_o(Fa),A2),B2) ) ).

% curryI
tff(fact_6228_curryI,axiom,
    ! [Fa: fun_Pr5555124106300582480_num_o,A2: nat,B2: num] :
      ( aa_Pro6908818167415475913_num_o(Fa,aa_num5026351239867140595at_num(product_Pair_nat_num(A2),B2))
     => aa_num_o(aa_nat_fun_num_o(produc156083480235303841_num_o(Fa),A2),B2) ) ).

% curryI
tff(fact_6229_curryI,axiom,
    ! [Fa: fun_Pr4796943543331071514_nat_o,A2: nat,B2: nat] :
      ( aa_Pro6150637604445964947_nat_o(Fa,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2))
     => aa_nat_o(aa_nat_fun_nat_o(produc1310100445399344235_nat_o(Fa),A2),B2) ) ).

% curryI
tff(fact_6230_curryI,axiom,
    ! [Fa: fun_Pr3238374608701176162_int_o,A2: int,B2: int] :
      ( aa_Pro4592068669816069595_int_o(Fa,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2))
     => aa_int_o(aa_int_fun_int_o(produc175634133007206835_int_o(Fa),A2),B2) ) ).

% curryI
tff(fact_6231__092_060open_062_092_060And_062xa_Atia_O_Arefines_A_Ivebt__predi_Atia_Axa_J_A_IHeap_OHeap_AMap_Oempty_J_092_060close_062,axiom,
    ! [Tiaa: vEBT_VEBTi,Xaa: nat] : refine7594492741263601813on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_predi,Tiaa),Xaa),heap_T5286843759275942675on_nat(aTP_Lamm_rp)) ).

% \<open>\<And>xa tia. refines (vebt_predi tia xa) (Heap.Heap Map.empty)\<close>
tff(fact_6232_add_Oinverse__neutral,axiom,
    uminus8244633308260627903l_num1(zero_z3563351764282998399l_num1) = zero_z3563351764282998399l_num1 ).

% add.inverse_neutral
tff(fact_6233_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_6234_add_Oinverse__neutral,axiom,
    uminus_uminus_uint32(zero_zero_uint32) = zero_zero_uint32 ).

% add.inverse_neutral
tff(fact_6235_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_6236_add_Oinverse__neutral,axiom,
    uminus_uminus_rat(zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_6237_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_6238_neg__0__equal__iff__equal,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( zero_z3563351764282998399l_num1 = uminus8244633308260627903l_num1(A2) )
    <=> ( zero_z3563351764282998399l_num1 = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_6239_neg__0__equal__iff__equal,axiom,
    ! [A2: complex] :
      ( ( zero_zero_complex = aa_complex_complex(uminus1482373934393186551omplex,A2) )
    <=> ( zero_zero_complex = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_6240_neg__0__equal__iff__equal,axiom,
    ! [A2: uint32] :
      ( ( zero_zero_uint32 = uminus_uminus_uint32(A2) )
    <=> ( zero_zero_uint32 = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_6241_neg__0__equal__iff__equal,axiom,
    ! [A2: real] :
      ( ( zero_zero_real = aa_real_real(uminus_uminus_real,A2) )
    <=> ( zero_zero_real = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_6242_neg__0__equal__iff__equal,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat = uminus_uminus_rat(A2) )
    <=> ( zero_zero_rat = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_6243_neg__0__equal__iff__equal,axiom,
    ! [A2: int] :
      ( ( zero_zero_int = aa_int_int(uminus_uminus_int,A2) )
    <=> ( zero_zero_int = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_6244_neg__equal__0__iff__equal,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( ( uminus8244633308260627903l_num1(A2) = zero_z3563351764282998399l_num1 )
    <=> ( A2 = zero_z3563351764282998399l_num1 ) ) ).

% neg_equal_0_iff_equal
tff(fact_6245_neg__equal__0__iff__equal,axiom,
    ! [A2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A2) = zero_zero_complex )
    <=> ( A2 = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
tff(fact_6246_neg__equal__0__iff__equal,axiom,
    ! [A2: uint32] :
      ( ( uminus_uminus_uint32(A2) = zero_zero_uint32 )
    <=> ( A2 = zero_zero_uint32 ) ) ).

% neg_equal_0_iff_equal
tff(fact_6247_neg__equal__0__iff__equal,axiom,
    ! [A2: real] :
      ( ( aa_real_real(uminus_uminus_real,A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
tff(fact_6248_neg__equal__0__iff__equal,axiom,
    ! [A2: rat] :
      ( ( uminus_uminus_rat(A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_6249_neg__equal__0__iff__equal,axiom,
    ! [A2: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
tff(fact_6250_equal__neg__zero,axiom,
    ! [A2: real] :
      ( ( A2 = aa_real_real(uminus_uminus_real,A2) )
    <=> ( A2 = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_6251_equal__neg__zero,axiom,
    ! [A2: rat] :
      ( ( A2 = uminus_uminus_rat(A2) )
    <=> ( A2 = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_6252_equal__neg__zero,axiom,
    ! [A2: int] :
      ( ( A2 = aa_int_int(uminus_uminus_int,A2) )
    <=> ( A2 = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_6253_neg__equal__zero,axiom,
    ! [A2: real] :
      ( ( aa_real_real(uminus_uminus_real,A2) = A2 )
    <=> ( A2 = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_6254_neg__equal__zero,axiom,
    ! [A2: rat] :
      ( ( uminus_uminus_rat(A2) = A2 )
    <=> ( A2 = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_6255_neg__equal__zero,axiom,
    ! [A2: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = A2 )
    <=> ( A2 = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_6256_neg__le__iff__le,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% neg_le_iff_le
tff(fact_6257_neg__le__iff__le,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B2)),uminus_uminus_rat(A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% neg_le_iff_le
tff(fact_6258_neg__le__iff__le,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% neg_le_iff_le
tff(fact_6259_neg__less__iff__less,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% neg_less_iff_less
tff(fact_6260_neg__less__iff__less,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B2)),uminus_uminus_rat(A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% neg_less_iff_less
tff(fact_6261_neg__less__iff__less,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% neg_less_iff_less
tff(fact_6262_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_6263_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(M)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_6264_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(M)) = uminus_uminus_rat(numeral_numeral_rat(N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_6265_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
    <=> ( M = N ) ) ).

% neg_numeral_eq_iff
tff(fact_6266_mult__minus__left,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),B2) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2)) ).

% mult_minus_left
tff(fact_6267_mult__minus__left,axiom,
    ! [A2: uint32,B2: uint32] : aa_uint32_uint32(times_times_uint32(uminus_uminus_uint32(A2)),B2) = uminus_uminus_uint32(aa_uint32_uint32(times_times_uint32(A2),B2)) ).

% mult_minus_left
tff(fact_6268_mult__minus__left,axiom,
    ! [A2: real,B2: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A2)),B2) = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(A2),B2)) ).

% mult_minus_left
tff(fact_6269_mult__minus__left,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A2)),B2) = uminus_uminus_rat(aa_rat_rat(times_times_rat(A2),B2)) ).

% mult_minus_left
tff(fact_6270_mult__minus__left,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A2)),B2) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) ).

% mult_minus_left
tff(fact_6271_minus__mult__minus,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2) ).

% minus_mult_minus
tff(fact_6272_minus__mult__minus,axiom,
    ! [A2: uint32,B2: uint32] : aa_uint32_uint32(times_times_uint32(uminus_uminus_uint32(A2)),uminus_uminus_uint32(B2)) = aa_uint32_uint32(times_times_uint32(A2),B2) ).

% minus_mult_minus
tff(fact_6273_minus__mult__minus,axiom,
    ! [A2: real,B2: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A2)),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(times_times_real(A2),B2) ).

% minus_mult_minus
tff(fact_6274_minus__mult__minus,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A2)),uminus_uminus_rat(B2)) = aa_rat_rat(times_times_rat(A2),B2) ).

% minus_mult_minus
tff(fact_6275_minus__mult__minus,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A2)),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2) ).

% minus_mult_minus
tff(fact_6276_mult__minus__right,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),B2)) ).

% mult_minus_right
tff(fact_6277_mult__minus__right,axiom,
    ! [A2: uint32,B2: uint32] : aa_uint32_uint32(times_times_uint32(A2),uminus_uminus_uint32(B2)) = uminus_uminus_uint32(aa_uint32_uint32(times_times_uint32(A2),B2)) ).

% mult_minus_right
tff(fact_6278_mult__minus__right,axiom,
    ! [A2: real,B2: real] : aa_real_real(times_times_real(A2),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(A2),B2)) ).

% mult_minus_right
tff(fact_6279_mult__minus__right,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(times_times_rat(A2),uminus_uminus_rat(B2)) = uminus_uminus_rat(aa_rat_rat(times_times_rat(A2),B2)) ).

% mult_minus_right
tff(fact_6280_mult__minus__right,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) ).

% mult_minus_right
tff(fact_6281_div__minus__minus,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A2)),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ).

% div_minus_minus
tff(fact_6282_dvd__minus__iff,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Xb),uminus8244633308260627903l_num1(Y))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Xb),Y) ) ).

% dvd_minus_iff
tff(fact_6283_dvd__minus__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_6284_dvd__minus__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( aa_complex_o(dvd_dvd_complex(Xb),aa_complex_complex(uminus1482373934393186551omplex,Y))
    <=> aa_complex_o(dvd_dvd_complex(Xb),Y) ) ).

% dvd_minus_iff
tff(fact_6285_dvd__minus__iff,axiom,
    ! [Xb: uint32,Y: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(Xb),uminus_uminus_uint32(Y))
    <=> aa_uint32_o(dvd_dvd_uint32(Xb),Y) ) ).

% dvd_minus_iff
tff(fact_6286_dvd__minus__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_real_real(uminus_uminus_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_6287_dvd__minus__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),uminus_uminus_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_6288_dvd__minus__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_int_int(uminus_uminus_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_6289_minus__dvd__iff,axiom,
    ! [Xb: word_N3645301735248828278l_num1,Y: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(uminus8244633308260627903l_num1(Xb)),Y)
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Xb),Y) ) ).

% minus_dvd_iff
tff(fact_6290_minus__dvd__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),Y)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_6291_minus__dvd__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),Y)
    <=> aa_complex_o(dvd_dvd_complex(Xb),Y) ) ).

% minus_dvd_iff
tff(fact_6292_minus__dvd__iff,axiom,
    ! [Xb: uint32,Y: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(uminus_uminus_uint32(Xb)),Y)
    <=> aa_uint32_o(dvd_dvd_uint32(Xb),Y) ) ).

% minus_dvd_iff
tff(fact_6293_minus__dvd__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(uminus_uminus_real,Xb)),Y)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_6294_minus__dvd__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,uminus_uminus_rat(Xb)),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_6295_minus__dvd__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(uminus_uminus_int,Xb)),Y)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_6296_mod__minus__minus,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2)) ).

% mod_minus_minus
tff(fact_6297_mod__minus__minus,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A2)),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)) ).

% mod_minus_minus
tff(fact_6298_neg__0__le__iff__le,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) ) ).

% neg_0_le_iff_le
tff(fact_6299_neg__0__le__iff__le,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_6300_neg__0__le__iff__le,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_6301_neg__le__0__iff__le,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ).

% neg_le_0_iff_le
tff(fact_6302_neg__le__0__iff__le,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A2)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ).

% neg_le_0_iff_le
tff(fact_6303_neg__le__0__iff__le,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ).

% neg_le_0_iff_le
tff(fact_6304_less__eq__neg__nonpos,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(uminus_uminus_real,A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) ) ).

% less_eq_neg_nonpos
tff(fact_6305_less__eq__neg__nonpos,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),uminus_uminus_rat(A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_6306_less__eq__neg__nonpos,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_6307_neg__less__eq__nonneg,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ).

% neg_less_eq_nonneg
tff(fact_6308_neg__less__eq__nonneg,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A2)),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ).

% neg_less_eq_nonneg
tff(fact_6309_neg__less__eq__nonneg,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),A2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ).

% neg_less_eq_nonneg
tff(fact_6310_less__neg__neg,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(uminus_uminus_real,A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real) ) ).

% less_neg_neg
tff(fact_6311_less__neg__neg,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),uminus_uminus_rat(A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_6312_less__neg__neg,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ).

% less_neg_neg
tff(fact_6313_neg__less__pos,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A2)),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2) ) ).

% neg_less_pos
tff(fact_6314_neg__less__pos,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(A2)),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ).

% neg_less_pos
tff(fact_6315_neg__less__pos,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A2)),A2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2) ) ).

% neg_less_pos
tff(fact_6316_neg__0__less__iff__less,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real) ) ).

% neg_0_less_iff_less
tff(fact_6317_neg__0__less__iff__less,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),uminus_uminus_rat(A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_6318_neg__0__less__iff__less,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_6319_neg__less__0__iff__less,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A2)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2) ) ).

% neg_less_0_iff_less
tff(fact_6320_neg__less__0__iff__less,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(A2)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ).

% neg_less_0_iff_less
tff(fact_6321_neg__less__0__iff__less,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2) ) ).

% neg_less_0_iff_less
tff(fact_6322_add_Oright__inverse,axiom,
    ! [A2: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(A2,uminus8244633308260627903l_num1(A2)) = zero_z3563351764282998399l_num1 ).

% add.right_inverse
tff(fact_6323_add_Oright__inverse,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),aa_complex_complex(uminus1482373934393186551omplex,A2)) = zero_zero_complex ).

% add.right_inverse
tff(fact_6324_add_Oright__inverse,axiom,
    ! [A2: uint32] : plus_plus_uint32(A2,uminus_uminus_uint32(A2)) = zero_zero_uint32 ).

% add.right_inverse
tff(fact_6325_add_Oright__inverse,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(uminus_uminus_real,A2)) = zero_zero_real ).

% add.right_inverse
tff(fact_6326_add_Oright__inverse,axiom,
    ! [A2: rat] : plus_plus_rat(A2,uminus_uminus_rat(A2)) = zero_zero_rat ).

% add.right_inverse
tff(fact_6327_add_Oright__inverse,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_int_int(uminus_uminus_int,A2)) = zero_zero_int ).

% add.right_inverse
tff(fact_6328_ab__left__minus,axiom,
    ! [A2: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(uminus8244633308260627903l_num1(A2),A2) = zero_z3563351764282998399l_num1 ).

% ab_left_minus
tff(fact_6329_ab__left__minus,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),A2) = zero_zero_complex ).

% ab_left_minus
tff(fact_6330_ab__left__minus,axiom,
    ! [A2: uint32] : plus_plus_uint32(uminus_uminus_uint32(A2),A2) = zero_zero_uint32 ).

% ab_left_minus
tff(fact_6331_ab__left__minus,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),A2) = zero_zero_real ).

% ab_left_minus
tff(fact_6332_ab__left__minus,axiom,
    ! [A2: rat] : plus_plus_rat(uminus_uminus_rat(A2),A2) = zero_zero_rat ).

% ab_left_minus
tff(fact_6333_ab__left__minus,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A2)),A2) = zero_zero_int ).

% ab_left_minus
tff(fact_6334_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : plus_p361126936061061375l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(M)),uminus8244633308260627903l_num1(numera7442385471795722001l_num1(N))) = uminus8244633308260627903l_num1(plus_p361126936061061375l_num1(numera7442385471795722001l_num1(M),numera7442385471795722001l_num1(N))) ).

% add_neg_numeral_simps(3)
tff(fact_6335_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,numera6690914467698888265omplex(M)),numera6690914467698888265omplex(N))) ).

% add_neg_numeral_simps(3)
tff(fact_6336_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : plus_plus_uint32(uminus_uminus_uint32(numera9087168376688890119uint32(M)),uminus_uminus_uint32(numera9087168376688890119uint32(N))) = uminus_uminus_uint32(plus_plus_uint32(numera9087168376688890119uint32(M),numera9087168376688890119uint32(N))) ).

% add_neg_numeral_simps(3)
tff(fact_6337_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,numeral_numeral_real(M)),numeral_numeral_real(N))) ).

% add_neg_numeral_simps(3)
tff(fact_6338_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : plus_plus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(N))) = uminus_uminus_rat(plus_plus_rat(numeral_numeral_rat(M),numeral_numeral_rat(N))) ).

% add_neg_numeral_simps(3)
tff(fact_6339_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N))) ).

% add_neg_numeral_simps(3)
tff(fact_6340_diff__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] : minus_4019991460397169231l_num1(zero_z3563351764282998399l_num1,A2) = uminus8244633308260627903l_num1(A2) ).

% diff_0
tff(fact_6341_diff__0,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,zero_zero_complex),A2) = aa_complex_complex(uminus1482373934393186551omplex,A2) ).

% diff_0
tff(fact_6342_diff__0,axiom,
    ! [A2: uint32] : minus_minus_uint32(zero_zero_uint32,A2) = uminus_uminus_uint32(A2) ).

% diff_0
tff(fact_6343_diff__0,axiom,
    ! [A2: real] : aa_real_real(minus_minus_real(zero_zero_real),A2) = aa_real_real(uminus_uminus_real,A2) ).

% diff_0
tff(fact_6344_diff__0,axiom,
    ! [A2: rat] : minus_minus_rat(zero_zero_rat,A2) = uminus_uminus_rat(A2) ).

% diff_0
tff(fact_6345_diff__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),A2) = aa_int_int(uminus_uminus_int,A2) ).

% diff_0
tff(fact_6346_verit__minus__simplify_I3_J,axiom,
    ! [B2: word_N3645301735248828278l_num1] : minus_4019991460397169231l_num1(zero_z3563351764282998399l_num1,B2) = uminus8244633308260627903l_num1(B2) ).

% verit_minus_simplify(3)
tff(fact_6347_verit__minus__simplify_I3_J,axiom,
    ! [B2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,zero_zero_complex),B2) = aa_complex_complex(uminus1482373934393186551omplex,B2) ).

% verit_minus_simplify(3)
tff(fact_6348_verit__minus__simplify_I3_J,axiom,
    ! [B2: uint32] : minus_minus_uint32(zero_zero_uint32,B2) = uminus_uminus_uint32(B2) ).

% verit_minus_simplify(3)
tff(fact_6349_verit__minus__simplify_I3_J,axiom,
    ! [B2: real] : aa_real_real(minus_minus_real(zero_zero_real),B2) = aa_real_real(uminus_uminus_real,B2) ).

% verit_minus_simplify(3)
tff(fact_6350_verit__minus__simplify_I3_J,axiom,
    ! [B2: rat] : minus_minus_rat(zero_zero_rat,B2) = uminus_uminus_rat(B2) ).

% verit_minus_simplify(3)
tff(fact_6351_verit__minus__simplify_I3_J,axiom,
    ! [B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),B2) = aa_int_int(uminus_uminus_int,B2) ).

% verit_minus_simplify(3)
tff(fact_6352_mult__minus1,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Z) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1
tff(fact_6353_mult__minus1,axiom,
    ! [Z: uint32] : aa_uint32_uint32(times_times_uint32(uminus_uminus_uint32(one_one_uint32)),Z) = uminus_uminus_uint32(Z) ).

% mult_minus1
tff(fact_6354_mult__minus1,axiom,
    ! [Z: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,one_one_real)),Z) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1
tff(fact_6355_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(one_one_rat)),Z) = uminus_uminus_rat(Z) ).

% mult_minus1
tff(fact_6356_mult__minus1,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,one_one_int)),Z) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1
tff(fact_6357_mult__minus1__right,axiom,
    ! [Z: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Z),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1_right
tff(fact_6358_mult__minus1__right,axiom,
    ! [Z: uint32] : aa_uint32_uint32(times_times_uint32(Z),uminus_uminus_uint32(one_one_uint32)) = uminus_uminus_uint32(Z) ).

% mult_minus1_right
tff(fact_6359_mult__minus1__right,axiom,
    ! [Z: real] : aa_real_real(times_times_real(Z),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1_right
tff(fact_6360_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(Z) ).

% mult_minus1_right
tff(fact_6361_mult__minus1__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1_right
tff(fact_6362_divide__minus1,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Xb) ).

% divide_minus1
tff(fact_6363_divide__minus1,axiom,
    ! [Xb: real] : aa_real_real(divide_divide_real(Xb),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Xb) ).

% divide_minus1
tff(fact_6364_divide__minus1,axiom,
    ! [Xb: rat] : divide_divide_rat(Xb,uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(Xb) ).

% divide_minus1
tff(fact_6365_div__minus1__right,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A2) ).

% div_minus1_right
tff(fact_6366_minus__mod__self1,axiom,
    ! [B2: code_integer,A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B2),A2)),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B2) ).

% minus_mod_self1
tff(fact_6367_minus__mod__self1,axiom,
    ! [B2: int,A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A2)),B2) ).

% minus_mod_self1
tff(fact_6368_real__add__minus__iff,axiom,
    ! [Xb: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(uminus_uminus_real,A2)) = zero_zero_real )
    <=> ( Xb = A2 ) ) ).

% real_add_minus_iff
tff(fact_6369_add__neg__numeral__special_I8_J,axiom,
    plus_p361126936061061375l_num1(uminus8244633308260627903l_num1(one_on7727431528512463931l_num1),one_on7727431528512463931l_num1) = zero_z3563351764282998399l_num1 ).

% add_neg_numeral_special(8)
tff(fact_6370_add__neg__numeral__special_I8_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = zero_zero_complex ).

% add_neg_numeral_special(8)
tff(fact_6371_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)
tff(fact_6372_add__neg__numeral__special_I8_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = zero_zero_real ).

% add_neg_numeral_special(8)
tff(fact_6373_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)
tff(fact_6374_add__neg__numeral__special_I8_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = zero_zero_int ).

% add_neg_numeral_special(8)
tff(fact_6375_add__neg__numeral__special_I7_J,axiom,
    plus_p361126936061061375l_num1(one_on7727431528512463931l_num1,uminus8244633308260627903l_num1(one_on7727431528512463931l_num1)) = zero_z3563351764282998399l_num1 ).

% add_neg_numeral_special(7)
tff(fact_6376_add__neg__numeral__special_I7_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% add_neg_numeral_special(7)
tff(fact_6377_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)
tff(fact_6378_add__neg__numeral__special_I7_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% add_neg_numeral_special(7)
tff(fact_6379_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)
tff(fact_6380_add__neg__numeral__special_I7_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% add_neg_numeral_special(7)
tff(fact_6381_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6382_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6383_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( uminus_uminus_rat(one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6384_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_6385_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6386_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6387_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(N)) = uminus_uminus_rat(one_one_rat) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6388_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_6389_diff__numeral__special_I12_J,axiom,
    minus_4019991460397169231l_num1(uminus8244633308260627903l_num1(one_on7727431528512463931l_num1),uminus8244633308260627903l_num1(one_on7727431528512463931l_num1)) = zero_z3563351764282998399l_num1 ).

% diff_numeral_special(12)
tff(fact_6390_diff__numeral__special_I12_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% diff_numeral_special(12)
tff(fact_6391_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)
tff(fact_6392_diff__numeral__special_I12_J,axiom,
    aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% diff_numeral_special(12)
tff(fact_6393_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)
tff(fact_6394_diff__numeral__special_I12_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% diff_numeral_special(12)
tff(fact_6395_minus__one__mult__self,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_6396_minus__one__mult__self,axiom,
    ! [N: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)) = one_one_complex ).

% minus_one_mult_self
tff(fact_6397_minus__one__mult__self,axiom,
    ! [N: nat] : aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(one_one_uint32)),N)),aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(one_one_uint32)),N)) = one_one_uint32 ).

% minus_one_mult_self
tff(fact_6398_minus__one__mult__self,axiom,
    ! [N: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)) = one_one_real ).

% minus_one_mult_self
tff(fact_6399_minus__one__mult__self,axiom,
    ! [N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)) = one_one_rat ).

% minus_one_mult_self
tff(fact_6400_minus__one__mult__self,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)) = one_one_int ).

% minus_one_mult_self
tff(fact_6401_left__minus__one__mult__self,axiom,
    ! [N: nat,A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),A2)) = A2 ).

% left_minus_one_mult_self
tff(fact_6402_left__minus__one__mult__self,axiom,
    ! [N: nat,A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),A2)) = A2 ).

% left_minus_one_mult_self
tff(fact_6403_left__minus__one__mult__self,axiom,
    ! [N: nat,A2: uint32] : aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(one_one_uint32)),N)),aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(one_one_uint32)),N)),A2)) = A2 ).

% left_minus_one_mult_self
tff(fact_6404_left__minus__one__mult__self,axiom,
    ! [N: nat,A2: real] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),A2)) = A2 ).

% left_minus_one_mult_self
tff(fact_6405_left__minus__one__mult__self,axiom,
    ! [N: nat,A2: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),A2)) = A2 ).

% left_minus_one_mult_self
tff(fact_6406_left__minus__one__mult__self,axiom,
    ! [N: nat,A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),A2)) = A2 ).

% left_minus_one_mult_self
tff(fact_6407_mod__minus1__right,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_6408_mod__minus1__right,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_6409_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_Cod1783974652533396728nteger(ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))) ).

% max_number_of(4)
tff(fact_6410_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ord_ma8239519435860878689l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(U)),uminus8244633308260627903l_num1(numera7442385471795722001l_num1(V))) = $ite(aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(U))),uminus8244633308260627903l_num1(numera7442385471795722001l_num1(V))),uminus8244633308260627903l_num1(numera7442385471795722001l_num1(V)),uminus8244633308260627903l_num1(numera7442385471795722001l_num1(U))) ).

% max_number_of(4)
tff(fact_6411_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ord_max_uint32(uminus_uminus_uint32(numera9087168376688890119uint32(U)),uminus_uminus_uint32(numera9087168376688890119uint32(V))) = $ite(aa_uint32_o(ord_less_eq_uint32(uminus_uminus_uint32(numera9087168376688890119uint32(U))),uminus_uminus_uint32(numera9087168376688890119uint32(V))),uminus_uminus_uint32(numera9087168376688890119uint32(V)),uminus_uminus_uint32(numera9087168376688890119uint32(U))) ).

% max_number_of(4)
tff(fact_6412_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V)),aa_real_real(uminus_uminus_real,numeral_numeral_real(U))) ).

% max_number_of(4)
tff(fact_6413_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ord_max_rat(uminus_uminus_rat(numeral_numeral_rat(U)),uminus_uminus_rat(numeral_numeral_rat(V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(U))),uminus_uminus_rat(numeral_numeral_rat(V))),uminus_uminus_rat(numeral_numeral_rat(V)),uminus_uminus_rat(numeral_numeral_rat(U))) ).

% max_number_of(4)
tff(fact_6414_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(4)
tff(fact_6415_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_Cod1783974652533396728nteger(ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)),numera6620942414471956472nteger(V),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))) ).

% max_number_of(3)
tff(fact_6416_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ord_ma8239519435860878689l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(U)),numera7442385471795722001l_num1(V)) = $ite(aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(U))),numera7442385471795722001l_num1(V)),numera7442385471795722001l_num1(V),uminus8244633308260627903l_num1(numera7442385471795722001l_num1(U))) ).

% max_number_of(3)
tff(fact_6417_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ord_max_uint32(uminus_uminus_uint32(numera9087168376688890119uint32(U)),numera9087168376688890119uint32(V)) = $ite(aa_uint32_o(ord_less_eq_uint32(uminus_uminus_uint32(numera9087168376688890119uint32(U))),numera9087168376688890119uint32(V)),numera9087168376688890119uint32(V),uminus_uminus_uint32(numera9087168376688890119uint32(U))) ).

% max_number_of(3)
tff(fact_6418_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(U))),numeral_numeral_real(V)),numeral_numeral_real(V),aa_real_real(uminus_uminus_real,numeral_numeral_real(U))) ).

% max_number_of(3)
tff(fact_6419_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ord_max_rat(uminus_uminus_rat(numeral_numeral_rat(U)),numeral_numeral_rat(V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(U))),numeral_numeral_rat(V)),numeral_numeral_rat(V),uminus_uminus_rat(numeral_numeral_rat(U))) ).

% max_number_of(3)
tff(fact_6420_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(3)
tff(fact_6421_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_Cod1783974652533396728nteger(ord_max_Code_integer(numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),numera6620942414471956472nteger(U)) ).

% max_number_of(2)
tff(fact_6422_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ord_ma8239519435860878689l_num1(numera7442385471795722001l_num1(U),uminus8244633308260627903l_num1(numera7442385471795722001l_num1(V))) = $ite(aa_wor4452366487969921764num1_o(ord_le3335648743751981014l_num1(numera7442385471795722001l_num1(U)),uminus8244633308260627903l_num1(numera7442385471795722001l_num1(V))),uminus8244633308260627903l_num1(numera7442385471795722001l_num1(V)),numera7442385471795722001l_num1(U)) ).

% max_number_of(2)
tff(fact_6423_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ord_max_uint32(numera9087168376688890119uint32(U),uminus_uminus_uint32(numera9087168376688890119uint32(V))) = $ite(aa_uint32_o(ord_less_eq_uint32(numera9087168376688890119uint32(U)),uminus_uminus_uint32(numera9087168376688890119uint32(V))),uminus_uminus_uint32(numera9087168376688890119uint32(V)),numera9087168376688890119uint32(U)) ).

% max_number_of(2)
tff(fact_6424_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(ord_max_real(numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(U)),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(uminus_uminus_real,numeral_numeral_real(V)),numeral_numeral_real(U)) ).

% max_number_of(2)
tff(fact_6425_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ord_max_rat(numeral_numeral_rat(U),uminus_uminus_rat(numeral_numeral_rat(V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(U)),uminus_uminus_rat(numeral_numeral_rat(V))),uminus_uminus_rat(numeral_numeral_rat(V)),numeral_numeral_rat(U)) ).

% max_number_of(2)
tff(fact_6426_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(2)
tff(fact_6427_norm__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = numeral_numeral_real(W) ).

% norm_neg_numeral
tff(fact_6428_norm__neg__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = numeral_numeral_real(W) ).

% norm_neg_numeral
tff(fact_6429_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(V)),plus_p361126936061061375l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(W)),Y)) = plus_p361126936061061375l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Y) ).

% semiring_norm(168)
tff(fact_6430_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_6431_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(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Y) ).

% semiring_norm(168)
tff(fact_6432_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_6433_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(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W))),Y) ).

% semiring_norm(168)
tff(fact_6434_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W)))),Y) ).

% semiring_norm(168)
tff(fact_6435_ceiling__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_6436_ceiling__neg__numeral,axiom,
    ! [V: num] : archim2889992004027027881ng_rat(uminus_uminus_rat(numeral_numeral_rat(V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_6437_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : minus_4019991460397169231l_num1(numera7442385471795722001l_num1(M),uminus8244633308260627903l_num1(numera7442385471795722001l_num1(N))) = numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_6438_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_6439_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : minus_minus_uint32(numera9087168376688890119uint32(M),uminus_uminus_uint32(numera9087168376688890119uint32(N))) = numera9087168376688890119uint32(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_6440_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_real_real(minus_minus_real(numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_6441_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : minus_minus_rat(numeral_numeral_rat(M),uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_6442_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N)) ).

% diff_numeral_simps(2)
tff(fact_6443_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : minus_4019991460397169231l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(M)),numera7442385471795722001l_num1(N)) = uminus8244633308260627903l_num1(numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_6444_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_6445_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : minus_minus_uint32(uminus_uminus_uint32(numera9087168376688890119uint32(M)),numera9087168376688890119uint32(N)) = uminus_uminus_uint32(numera9087168376688890119uint32(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_6446_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(N)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_6447_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(M)),numeral_numeral_rat(N)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_6448_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),N))) ).

% diff_numeral_simps(3)
tff(fact_6449_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(V))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(W))),Y)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_6450_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_6451_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: uint32] : aa_uint32_uint32(times_times_uint32(uminus_uminus_uint32(numera9087168376688890119uint32(V))),aa_uint32_uint32(times_times_uint32(uminus_uminus_uint32(numera9087168376688890119uint32(W))),Y)) = aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_6452_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_6453_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(V))),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_6454_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W))),Y) ).

% semiring_norm(172)
tff(fact_6455_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(V)),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(W))),Y)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6456_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(V)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6457_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: uint32] : aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(V)),aa_uint32_uint32(times_times_uint32(uminus_uminus_uint32(numera9087168376688890119uint32(W))),Y)) = aa_uint32_uint32(times_times_uint32(uminus_uminus_uint32(numera9087168376688890119uint32(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6458_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(times_times_real(numeral_numeral_real(V)),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),Y)) = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6459_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(V)),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6460_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(171)
tff(fact_6461_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(V))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(W)),Y)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6462_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6463_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: uint32] : aa_uint32_uint32(times_times_uint32(uminus_uminus_uint32(numera9087168376688890119uint32(V))),aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(W)),Y)) = aa_uint32_uint32(times_times_uint32(uminus_uminus_uint32(numera9087168376688890119uint32(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6464_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),aa_real_real(times_times_real(numeral_numeral_real(W)),Y)) = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6465_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(V))),aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),Y)) = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6466_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W)))),Y) ).

% semiring_norm(170)
tff(fact_6467_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(M))),uminus8244633308260627903l_num1(numera7442385471795722001l_num1(N))) = numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_6468_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_6469_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_uint32_uint32(times_times_uint32(uminus_uminus_uint32(numera9087168376688890119uint32(M))),uminus_uminus_uint32(numera9087168376688890119uint32(N))) = numera9087168376688890119uint32(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_6470_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_6471_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_6472_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N)) ).

% mult_neg_numeral_simps(1)
tff(fact_6473_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(M))),numera7442385471795722001l_num1(N)) = uminus8244633308260627903l_num1(numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_6474_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(N)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_6475_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_uint32_uint32(times_times_uint32(uminus_uminus_uint32(numera9087168376688890119uint32(M))),numera9087168376688890119uint32(N)) = uminus_uminus_uint32(numera9087168376688890119uint32(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_6476_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(N)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_6477_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(N)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_6478_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(2)
tff(fact_6479_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(M)),uminus8244633308260627903l_num1(numera7442385471795722001l_num1(N))) = uminus8244633308260627903l_num1(numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_6480_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_6481_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(M)),uminus_uminus_uint32(numera9087168376688890119uint32(N))) = uminus_uminus_uint32(numera9087168376688890119uint32(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_6482_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_real_real(times_times_real(numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_6483_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_rat_rat(times_times_rat(numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(N))) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_6484_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),N))) ).

% mult_neg_numeral_simps(3)
tff(fact_6485_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N),M) ) ).

% neg_numeral_le_iff
tff(fact_6486_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(N)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N),M) ) ).

% neg_numeral_le_iff
tff(fact_6487_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,N),M) ) ).

% neg_numeral_le_iff
tff(fact_6488_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,numeral_numeral_real(N)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,N),M) ) ).

% neg_numeral_less_iff
tff(fact_6489_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(numeral_numeral_rat(N)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,N),M) ) ).

% neg_numeral_less_iff
tff(fact_6490_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,N),M) ) ).

% neg_numeral_less_iff
tff(fact_6491_round__neg__numeral,axiom,
    ! [N: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% round_neg_numeral
tff(fact_6492_round__neg__numeral,axiom,
    ! [N: num] : archim7778729529865785530nd_rat(uminus_uminus_rat(numeral_numeral_rat(N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% round_neg_numeral
tff(fact_6493_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_6494_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(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
tff(fact_6495_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_6496_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_6497_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(one_one_rat))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_6498_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_6499_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: complex,W: num,A2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = A2 )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,B2 = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),A2 = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_6500_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( ( aa_real_real(divide_divide_real(B2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = A2 )
    <=> $ite(aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) != zero_zero_real,B2 = aa_real_real(times_times_real(A2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),A2 = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_6501_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( divide_divide_rat(B2,uminus_uminus_rat(numeral_numeral_rat(W))) = A2 )
    <=> $ite(uminus_uminus_rat(numeral_numeral_rat(W)) != zero_zero_rat,B2 = aa_rat_rat(times_times_rat(A2),uminus_uminus_rat(numeral_numeral_rat(W))),A2 = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_6502_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A2: complex,B2: complex,W: num] :
      ( ( A2 = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = B2,A2 = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_6503_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( A2 = aa_real_real(divide_divide_real(B2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) )
    <=> $ite(aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) != zero_zero_real,aa_real_real(times_times_real(A2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = B2,A2 = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_6504_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( A2 = divide_divide_rat(B2,uminus_uminus_rat(numeral_numeral_rat(W))) )
    <=> $ite(uminus_uminus_rat(numeral_numeral_rat(W)) != zero_zero_rat,aa_rat_rat(times_times_rat(A2),uminus_uminus_rat(numeral_numeral_rat(W))) = B2,A2 = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_6505_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),B2) ) ).

% divide_le_eq_numeral1(2)
tff(fact_6506_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B2,uminus_uminus_rat(numeral_numeral_rat(W)))),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),uminus_uminus_rat(numeral_numeral_rat(W)))),B2) ) ).

% divide_le_eq_numeral1(2)
tff(fact_6507_le__divide__eq__numeral1_I2_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(divide_divide_real(B2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(times_times_real(A2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_6508_le__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),divide_divide_rat(B2,uminus_uminus_rat(numeral_numeral_rat(W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_6509_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))),B2) ) ).

% divide_less_eq_numeral1(2)
tff(fact_6510_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B2,uminus_uminus_rat(numeral_numeral_rat(W)))),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),uminus_uminus_rat(numeral_numeral_rat(W)))),B2) ) ).

% divide_less_eq_numeral1(2)
tff(fact_6511_less__divide__eq__numeral1_I2_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(divide_divide_real(B2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(times_times_real(A2),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_6512_less__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),divide_divide_rat(B2,uminus_uminus_rat(numeral_numeral_rat(W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_6513_power2__minus,axiom,
    ! [A2: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_6514_power2__minus,axiom,
    ! [A2: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_6515_power2__minus,axiom,
    ! [A2: uint32] : aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_uint32(power_power_uint32(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_6516_power2__minus,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_6517_power2__minus,axiom,
    ! [A2: rat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_6518_power2__minus,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_6519_add__neg__numeral__special_I9_J,axiom,
    plus_p361126936061061375l_num1(uminus8244633308260627903l_num1(one_on7727431528512463931l_num1),uminus8244633308260627903l_num1(one_on7727431528512463931l_num1)) = uminus8244633308260627903l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_6520_add__neg__numeral__special_I9_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_6521_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(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_6522_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_6523_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(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_6524_add__neg__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_6525_diff__numeral__special_I11_J,axiom,
    minus_4019991460397169231l_num1(one_on7727431528512463931l_num1,uminus8244633308260627903l_num1(one_on7727431528512463931l_num1)) = numera7442385471795722001l_num1(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_6526_diff__numeral__special_I11_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_6527_diff__numeral__special_I11_J,axiom,
    minus_minus_uint32(one_one_uint32,uminus_uminus_uint32(one_one_uint32)) = numera9087168376688890119uint32(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_6528_diff__numeral__special_I11_J,axiom,
    aa_real_real(minus_minus_real(one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_6529_diff__numeral__special_I11_J,axiom,
    minus_minus_rat(one_one_rat,uminus_uminus_rat(one_one_rat)) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_6530_diff__numeral__special_I11_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_6531_diff__numeral__special_I10_J,axiom,
    minus_4019991460397169231l_num1(uminus8244633308260627903l_num1(one_on7727431528512463931l_num1),one_on7727431528512463931l_num1) = uminus8244633308260627903l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_6532_diff__numeral__special_I10_J,axiom,
    aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_6533_diff__numeral__special_I10_J,axiom,
    minus_minus_uint32(uminus_uminus_uint32(one_one_uint32),one_one_uint32) = uminus_uminus_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_6534_diff__numeral__special_I10_J,axiom,
    aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_6535_diff__numeral__special_I10_J,axiom,
    minus_minus_rat(uminus_uminus_rat(one_one_rat),one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_6536_diff__numeral__special_I10_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_6537_minus__1__div__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_6538_minus__1__mod__2__eq,axiom,
    aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_6539_minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% minus_1_mod_2_eq
tff(fact_6540_bits__minus__1__mod__2__eq,axiom,
    modulo1504961113040953224l_num1(uminus8244633308260627903l_num1(one_on7727431528512463931l_num1),numera7442385471795722001l_num1(aa_num_num(bit0,one))) = one_on7727431528512463931l_num1 ).

% bits_minus_1_mod_2_eq
tff(fact_6541_bits__minus__1__mod__2__eq,axiom,
    aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_minus_1_mod_2_eq
tff(fact_6542_bits__minus__1__mod__2__eq,axiom,
    aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,uminus_uminus_uint32(one_one_uint32)),numera9087168376688890119uint32(aa_num_num(bit0,one))) = one_one_uint32 ).

% bits_minus_1_mod_2_eq
tff(fact_6543_bits__minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_minus_1_mod_2_eq
tff(fact_6544_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6545_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6546_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: uint32,N: nat] : aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_uint32(power_power_uint32(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6547_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6548_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: rat,N: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6549_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% Power.ring_1_class.power_minus_even
tff(fact_6550_power__minus__odd,axiom,
    ! [N: nat,A2: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ) ) ).

% power_minus_odd
tff(fact_6551_power__minus__odd,axiom,
    ! [N: nat,A2: complex] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),N) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)) ) ) ).

% power_minus_odd
tff(fact_6552_power__minus__odd,axiom,
    ! [N: nat,A2: uint32] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(A2)),N) = uminus_uminus_uint32(aa_nat_uint32(power_power_uint32(A2),N)) ) ) ).

% power_minus_odd
tff(fact_6553_power__minus__odd,axiom,
    ! [N: nat,A2: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),N) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ) ) ).

% power_minus_odd
tff(fact_6554_power__minus__odd,axiom,
    ! [N: nat,A2: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A2)),N) = uminus_uminus_rat(aa_nat_rat(power_power_rat(A2),N)) ) ) ).

% power_minus_odd
tff(fact_6555_power__minus__odd,axiom,
    ! [N: nat,A2: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),N) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A2),N)) ) ) ).

% power_minus_odd
tff(fact_6556_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),N) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6557_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A2: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),N) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6558_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A2: uint32] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(A2)),N) = aa_nat_uint32(power_power_uint32(A2),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6559_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6560_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(A2)),N) = aa_nat_rat(power_power_rat(A2),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6561_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),N) = aa_nat_int(power_power_int(A2),N) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_6562_diff__numeral__special_I4_J,axiom,
    ! [M: num] : minus_4019991460397169231l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(M)),one_on7727431528512463931l_num1) = uminus8244633308260627903l_num1(numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_6563_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_6564_diff__numeral__special_I4_J,axiom,
    ! [M: num] : minus_minus_uint32(uminus_uminus_uint32(numera9087168376688890119uint32(M)),one_one_uint32) = uminus_uminus_uint32(numera9087168376688890119uint32(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_6565_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_6566_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(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_6567_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_6568_diff__numeral__special_I3_J,axiom,
    ! [N: num] : minus_4019991460397169231l_num1(one_on7727431528512463931l_num1,uminus8244633308260627903l_num1(numera7442385471795722001l_num1(N))) = numera7442385471795722001l_num1(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_6569_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_6570_diff__numeral__special_I3_J,axiom,
    ! [N: num] : minus_minus_uint32(one_one_uint32,uminus_uminus_uint32(numera9087168376688890119uint32(N))) = numera9087168376688890119uint32(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_6571_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_real_real(minus_minus_real(one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = numeral_numeral_real(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_6572_diff__numeral__special_I3_J,axiom,
    ! [N: num] : minus_minus_rat(one_one_rat,uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_6573_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),N)) ).

% diff_numeral_special(3)
tff(fact_6574_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),N) = aa_int_Code_integer(ring_18347121197199848620nteger,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6575_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: int] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xb))),N) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6576_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),N) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6577_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xb))),N) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6578_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_6579_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,N: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Y) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6580_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,N: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xb))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6581_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,N: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6582_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,N: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xb))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6583_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,N: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_6584_ceiling__le__neg__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(uminus_uminus_real,numeral_numeral_real(V))) ) ).

% ceiling_le_neg_numeral
tff(fact_6585_ceiling__le__neg__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,archim2889992004027027881ng_rat(Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),uminus_uminus_rat(numeral_numeral_rat(V))) ) ).

% ceiling_le_neg_numeral
tff(fact_6586_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),Xb) ) ).

% neg_numeral_less_ceiling
tff(fact_6587_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(V))),Xb) ) ).

% neg_numeral_less_ceiling
tff(fact_6588_ceiling__less__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% ceiling_less_zero
tff(fact_6589_ceiling__less__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xb)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),uminus_uminus_rat(one_one_rat)) ) ).

% ceiling_less_zero
tff(fact_6590_zero__le__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb) ) ).

% zero_le_ceiling
tff(fact_6591_zero__le__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),Xb) ) ).

% zero_le_ceiling
tff(fact_6592_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_6593_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_complex ).

% power_minus1_even
tff(fact_6594_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(one_one_uint32)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_uint32 ).

% power_minus1_even
tff(fact_6595_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_real ).

% power_minus1_even
tff(fact_6596_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_rat ).

% power_minus1_even
tff(fact_6597_power__minus1__even,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = one_one_int ).

% power_minus1_even
tff(fact_6598_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_6599_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% neg_one_odd_power
tff(fact_6600_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(one_one_uint32)),N) = uminus_uminus_uint32(one_one_uint32) ) ) ).

% neg_one_odd_power
tff(fact_6601_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% neg_one_odd_power
tff(fact_6602_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N) = uminus_uminus_rat(one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_6603_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_6604_neg__one__even__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_6605_neg__one__even__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = one_one_complex ) ) ).

% neg_one_even_power
tff(fact_6606_neg__one__even__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(one_one_uint32)),N) = one_one_uint32 ) ) ).

% neg_one_even_power
tff(fact_6607_neg__one__even__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) = one_one_real ) ) ).

% neg_one_even_power
tff(fact_6608_neg__one__even__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_6609_neg__one__even__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_6610_ceiling__less__neg__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)) ) ).

% ceiling_less_neg_numeral
tff(fact_6611_ceiling__less__neg__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,archim2889992004027027881ng_rat(Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_6612_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(V))),one_one_real)),Xb) ) ).

% neg_numeral_le_ceiling
tff(fact_6613_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),archim2889992004027027881ng_rat(Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(uminus_uminus_rat(numeral_numeral_rat(V)),one_one_rat)),Xb) ) ).

% neg_numeral_le_ceiling
tff(fact_6614_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)),A2) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6615_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),N)),aa_int_real(ring_1_of_int_real,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)),A2) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6616_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xb))),N)),aa_int_rat(ring_1_of_int_rat,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)),A2) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6617_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)),aa_int_int(ring_1_of_int_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)),A2) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6618_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A2)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6619_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6620_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A2)),aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xb))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6621_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A2)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6622_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),N)),aa_int_Code_integer(ring_18347121197199848620nteger,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)),A2) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6623_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),N)),aa_int_real(ring_1_of_int_real,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)),A2) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6624_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xb))),N)),aa_int_rat(ring_1_of_int_rat,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)),A2) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6625_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)),aa_int_int(ring_1_of_int_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)),A2) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6626_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,A2)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6627_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(Xb))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6628_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A2)),aa_nat_rat(power_power_rat(uminus_uminus_rat(numeral_numeral_rat(Xb))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6629_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A2)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),N)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6630_Heap__lub__empty,axiom,
    aa_set4319781776900185948on_nat(heap_T7048022066654196708on_nat,bot_bo8932748503833948152on_nat) = heap_T5286843759275942675on_nat(aTP_Lamm_rp) ).

% Heap_lub_empty
tff(fact_6631_Heap__lub__empty,axiom,
    aa_set2179536208438818360Heap_o(heap_Time_Heap_lub_o,bot_bo3236126332025433324Heap_o) = heap_Time_Heap_o2(aTP_Lamm_so) ).

% Heap_lub_empty
tff(fact_6632_Heap__lub__empty,axiom,
    aa_set8609060500495870620_VEBTi(heap_T3112222404744780921_VEBTi,bot_bo3125955617464001165_VEBTi) = heap_T1489671670754571048_VEBTi(aTP_Lamm_sp) ).

% Heap_lub_empty
tff(fact_6633_le__minus__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(uminus_uminus_real,B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(uminus_uminus_real,A2)) ) ).

% le_minus_iff
tff(fact_6634_le__minus__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),uminus_uminus_rat(B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),uminus_uminus_rat(A2)) ) ).

% le_minus_iff
tff(fact_6635_le__minus__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(uminus_uminus_int,B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(uminus_uminus_int,A2)) ) ).

% le_minus_iff
tff(fact_6636_minus__le__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),A2) ) ).

% minus_le_iff
tff(fact_6637_minus__le__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A2)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B2)),A2) ) ).

% minus_le_iff
tff(fact_6638_minus__le__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B2)),A2) ) ).

% minus_le_iff
tff(fact_6639_le__imp__neg__le,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A2)) ) ).

% le_imp_neg_le
tff(fact_6640_le__imp__neg__le,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B2)),uminus_uminus_rat(A2)) ) ).

% le_imp_neg_le
tff(fact_6641_le__imp__neg__le,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A2)) ) ).

% le_imp_neg_le
tff(fact_6642_less__minus__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(uminus_uminus_real,B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(uminus_uminus_real,A2)) ) ).

% less_minus_iff
tff(fact_6643_less__minus__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),uminus_uminus_rat(B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),uminus_uminus_rat(A2)) ) ).

% less_minus_iff
tff(fact_6644_less__minus__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(uminus_uminus_int,B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(uminus_uminus_int,A2)) ) ).

% less_minus_iff
tff(fact_6645_minus__less__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A2)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),A2) ) ).

% minus_less_iff
tff(fact_6646_minus__less__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(A2)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B2)),A2) ) ).

% minus_less_iff
tff(fact_6647_minus__less__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A2)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B2)),A2) ) ).

% minus_less_iff
tff(fact_6648_verit__negate__coefficient_I2_J,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A2)) ) ).

% verit_negate_coefficient(2)
tff(fact_6649_verit__negate__coefficient_I2_J,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B2)),uminus_uminus_rat(A2)) ) ).

% verit_negate_coefficient(2)
tff(fact_6650_verit__negate__coefficient_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A2)) ) ).

% verit_negate_coefficient(2)
tff(fact_6651_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) != numera6690914467698888265omplex(N) ).

% neg_numeral_neq_numeral
tff(fact_6652_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_real_real(uminus_uminus_real,numeral_numeral_real(M)) != numeral_numeral_real(N) ).

% neg_numeral_neq_numeral
tff(fact_6653_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : uminus_uminus_rat(numeral_numeral_rat(M)) != numeral_numeral_rat(N) ).

% neg_numeral_neq_numeral
tff(fact_6654_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) != aa_num_int(numeral_numeral_int,N) ).

% neg_numeral_neq_numeral
tff(fact_6655_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : numera6690914467698888265omplex(M) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% numeral_neq_neg_numeral
tff(fact_6656_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : numeral_numeral_real(M) != aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) ).

% numeral_neq_neg_numeral
tff(fact_6657_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : numeral_numeral_rat(M) != uminus_uminus_rat(numeral_numeral_rat(N)) ).

% numeral_neq_neg_numeral
tff(fact_6658_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] : aa_num_int(numeral_numeral_int,M) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% numeral_neq_neg_numeral
tff(fact_6659_is__num__normalize_I8_J,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,B2)),aa_complex_complex(uminus1482373934393186551omplex,A2)) ).

% is_num_normalize(8)
tff(fact_6660_is__num__normalize_I8_J,axiom,
    ! [A2: uint32,B2: uint32] : uminus_uminus_uint32(plus_plus_uint32(A2,B2)) = plus_plus_uint32(uminus_uminus_uint32(B2),uminus_uminus_uint32(A2)) ).

% is_num_normalize(8)
tff(fact_6661_is__num__normalize_I8_J,axiom,
    ! [A2: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A2)) ).

% is_num_normalize(8)
tff(fact_6662_is__num__normalize_I8_J,axiom,
    ! [A2: rat,B2: rat] : uminus_uminus_rat(plus_plus_rat(A2,B2)) = plus_plus_rat(uminus_uminus_rat(B2),uminus_uminus_rat(A2)) ).

% is_num_normalize(8)
tff(fact_6663_is__num__normalize_I8_J,axiom,
    ! [A2: int,B2: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A2)) ).

% is_num_normalize(8)
tff(fact_6664_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_6665_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_6666_one__neq__neg__one,axiom,
    one_one_rat != uminus_uminus_rat(one_one_rat) ).

% one_neq_neg_one
tff(fact_6667_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_6668_square__eq__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),A2) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B2),B2) )
    <=> ( ( A2 = B2 )
        | ( A2 = aa_complex_complex(uminus1482373934393186551omplex,B2) ) ) ) ).

% square_eq_iff
tff(fact_6669_square__eq__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(times_times_real(A2),A2) = aa_real_real(times_times_real(B2),B2) )
    <=> ( ( A2 = B2 )
        | ( A2 = aa_real_real(uminus_uminus_real,B2) ) ) ) ).

% square_eq_iff
tff(fact_6670_square__eq__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),A2) = aa_rat_rat(times_times_rat(B2),B2) )
    <=> ( ( A2 = B2 )
        | ( A2 = uminus_uminus_rat(B2) ) ) ) ).

% square_eq_iff
tff(fact_6671_square__eq__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A2),A2) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),B2) )
    <=> ( ( A2 = B2 )
        | ( A2 = aa_int_int(uminus_uminus_int,B2) ) ) ) ).

% square_eq_iff
tff(fact_6672_minus__mult__commute,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),B2) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% minus_mult_commute
tff(fact_6673_minus__mult__commute,axiom,
    ! [A2: uint32,B2: uint32] : aa_uint32_uint32(times_times_uint32(uminus_uminus_uint32(A2)),B2) = aa_uint32_uint32(times_times_uint32(A2),uminus_uminus_uint32(B2)) ).

% minus_mult_commute
tff(fact_6674_minus__mult__commute,axiom,
    ! [A2: real,B2: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A2)),B2) = aa_real_real(times_times_real(A2),aa_real_real(uminus_uminus_real,B2)) ).

% minus_mult_commute
tff(fact_6675_minus__mult__commute,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(A2)),B2) = aa_rat_rat(times_times_rat(A2),uminus_uminus_rat(B2)) ).

% minus_mult_commute
tff(fact_6676_minus__mult__commute,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A2)),B2) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),aa_int_int(uminus_uminus_int,B2)) ).

% minus_mult_commute
tff(fact_6677_minus__divide__right,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% minus_divide_right
tff(fact_6678_minus__divide__right,axiom,
    ! [A2: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A2),B2)) = aa_real_real(divide_divide_real(A2),aa_real_real(uminus_uminus_real,B2)) ).

% minus_divide_right
tff(fact_6679_minus__divide__right,axiom,
    ! [A2: rat,B2: rat] : uminus_uminus_rat(divide_divide_rat(A2,B2)) = divide_divide_rat(A2,uminus_uminus_rat(B2)) ).

% minus_divide_right
tff(fact_6680_minus__divide__divide,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2) ).

% minus_divide_divide
tff(fact_6681_minus__divide__divide,axiom,
    ! [A2: real,B2: real] : aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A2)),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(divide_divide_real(A2),B2) ).

% minus_divide_divide
tff(fact_6682_minus__divide__divide,axiom,
    ! [A2: rat,B2: rat] : divide_divide_rat(uminus_uminus_rat(A2),uminus_uminus_rat(B2)) = divide_divide_rat(A2,B2) ).

% minus_divide_divide
tff(fact_6683_minus__divide__left,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(uminus1482373934393186551omplex,A2)),B2) ).

% minus_divide_left
tff(fact_6684_minus__divide__left,axiom,
    ! [A2: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A2),B2)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A2)),B2) ).

% minus_divide_left
tff(fact_6685_minus__divide__left,axiom,
    ! [A2: rat,B2: rat] : uminus_uminus_rat(divide_divide_rat(A2,B2)) = divide_divide_rat(uminus_uminus_rat(A2),B2) ).

% minus_divide_left
tff(fact_6686_div__minus__right,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A2)),B2) ).

% div_minus_right
tff(fact_6687_mod__minus__eq,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2))),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B2) ).

% mod_minus_eq
tff(fact_6688_mod__minus__eq,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2))),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A2)),B2) ).

% mod_minus_eq
tff(fact_6689_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A2: code_integer,B2: code_integer,A4: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A4),B2) )
     => ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B2) = aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A4)),B2) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
tff(fact_6690_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A2: int,B2: int,A4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),B2) )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A4)),B2) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
tff(fact_6691_mod__minus__right,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B2)) ).

% mod_minus_right
tff(fact_6692_mod__minus__right,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A2)),B2)) ).

% mod_minus_right
tff(fact_6693_of__int__neg__numeral,axiom,
    ! [K: num] : ring_17408606157368542149l_num1(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = uminus8244633308260627903l_num1(numera7442385471795722001l_num1(K)) ).

% of_int_neg_numeral
tff(fact_6694_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K)) ).

% of_int_neg_numeral
tff(fact_6695_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_uint32(ring_1_of_int_uint32,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = uminus_uminus_uint32(numera9087168376688890119uint32(K)) ).

% of_int_neg_numeral
tff(fact_6696_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(K)) ).

% of_int_neg_numeral
tff(fact_6697_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = uminus_uminus_rat(numeral_numeral_rat(K)) ).

% of_int_neg_numeral
tff(fact_6698_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)) ).

% of_int_neg_numeral
tff(fact_6699_curryD,axiom,
    ! [Fa: fun_Pr6331313853612757382_num_o,A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(produc932273227547478743_num_o(Fa),A2),B2)
     => aa_Pro7685007914727650815_num_o(Fa,aa_num3156681643174609981um_num(product_Pair_num_num(A2),B2)) ) ).

% curryD
tff(fact_6700_curryD,axiom,
    ! [Fa: fun_Pr6320724876807202603VEBT_o,A2: nat,B2: produc4813437837504472865T_VEBT] :
      ( aa_Pro3461009536670225267VEBT_o(aa_nat6863640065291604488VEBT_o(produc1960320730580999890VEBT_o(Fa),A2),B2)
     => aa_Pro1497393596385541012VEBT_o(Fa,aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A2),B2)) ) ).

% curryD
tff(fact_6701_curryD,axiom,
    ! [Fa: fun_Pr5555124106300582480_num_o,A2: nat,B2: num] :
      ( aa_num_o(aa_nat_fun_num_o(produc156083480235303841_num_o(Fa),A2),B2)
     => aa_Pro6908818167415475913_num_o(Fa,aa_num5026351239867140595at_num(product_Pair_nat_num(A2),B2)) ) ).

% curryD
tff(fact_6702_curryD,axiom,
    ! [Fa: fun_Pr4796943543331071514_nat_o,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(produc1310100445399344235_nat_o(Fa),A2),B2)
     => aa_Pro6150637604445964947_nat_o(Fa,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2)) ) ).

% curryD
tff(fact_6703_curryD,axiom,
    ! [Fa: fun_Pr3238374608701176162_int_o,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(produc175634133007206835_int_o(Fa),A2),B2)
     => aa_Pro4592068669816069595_int_o(Fa,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2)) ) ).

% curryD
tff(fact_6704_curryE,axiom,
    ! [Fa: fun_Pr6331313853612757382_num_o,A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(produc932273227547478743_num_o(Fa),A2),B2)
     => aa_Pro7685007914727650815_num_o(Fa,aa_num3156681643174609981um_num(product_Pair_num_num(A2),B2)) ) ).

% curryE
tff(fact_6705_curryE,axiom,
    ! [Fa: fun_Pr6320724876807202603VEBT_o,A2: nat,B2: produc4813437837504472865T_VEBT] :
      ( aa_Pro3461009536670225267VEBT_o(aa_nat6863640065291604488VEBT_o(produc1960320730580999890VEBT_o(Fa),A2),B2)
     => aa_Pro1497393596385541012VEBT_o(Fa,aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(A2),B2)) ) ).

% curryE
tff(fact_6706_curryE,axiom,
    ! [Fa: fun_Pr5555124106300582480_num_o,A2: nat,B2: num] :
      ( aa_num_o(aa_nat_fun_num_o(produc156083480235303841_num_o(Fa),A2),B2)
     => aa_Pro6908818167415475913_num_o(Fa,aa_num5026351239867140595at_num(product_Pair_nat_num(A2),B2)) ) ).

% curryE
tff(fact_6707_curryE,axiom,
    ! [Fa: fun_Pr4796943543331071514_nat_o,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(produc1310100445399344235_nat_o(Fa),A2),B2)
     => aa_Pro6150637604445964947_nat_o(Fa,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A2),B2)) ) ).

% curryE
tff(fact_6708_curryE,axiom,
    ! [Fa: fun_Pr3238374608701176162_int_o,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(produc175634133007206835_int_o(Fa),A2),B2)
     => aa_Pro4592068669816069595_int_o(Fa,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2)) ) ).

% curryE
tff(fact_6709_TBOUND__empty,axiom,
    ! [Tb: nat] : aa_nat_o(time_TBOUND_nat(heap_Time_Heap_nat2(aTP_Lamm_sq)),Tb) ).

% TBOUND_empty
tff(fact_6710_TBOUND__empty,axiom,
    ! [Tb: nat] : aa_nat_o(time_T8353473612707095248on_nat(heap_T5286843759275942675on_nat(aTP_Lamm_rp)),Tb) ).

% TBOUND_empty
tff(fact_6711_TBOUND__empty,axiom,
    ! [Tb: nat] : aa_nat_o(time_TBOUND_o(heap_Time_Heap_o2(aTP_Lamm_so)),Tb) ).

% TBOUND_empty
tff(fact_6712_TBOUND__empty,axiom,
    ! [Tb: nat] : aa_nat_o(time_T5737551269749752165_VEBTi(heap_T1489671670754571048_VEBTi(aTP_Lamm_sp)),Tb) ).

% TBOUND_empty
tff(fact_6713_refines__empty,axiom,
    ! [M: heap_T2636463487746394924on_nat] : refine7594492741263601813on_nat(M,heap_T5286843759275942675on_nat(aTP_Lamm_rp)) ).

% refines_empty
tff(fact_6714_refines__empty,axiom,
    ! [M: heap_Time_Heap_o] : aa_Hea4522166450600650424ap_o_o(refine_Imp_refines_o(M),heap_Time_Heap_o2(aTP_Lamm_so)) ).

% refines_empty
tff(fact_6715_refines__empty,axiom,
    ! [M: heap_T8145700208782473153_VEBTi] : refine5565527176597971370_VEBTi(M,heap_T1489671670754571048_VEBTi(aTP_Lamm_sp)) ).

% refines_empty
tff(fact_6716_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% zero_neq_neg_numeral
tff(fact_6717_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_real != aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) ).

% zero_neq_neg_numeral
tff(fact_6718_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_rat != uminus_uminus_rat(numeral_numeral_rat(N)) ).

% zero_neq_neg_numeral
tff(fact_6719_zero__neq__neg__numeral,axiom,
    ! [N: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% zero_neq_neg_numeral
tff(fact_6720_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(N)) ).

% neg_numeral_le_numeral
tff(fact_6721_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(N)) ).

% neg_numeral_le_numeral
tff(fact_6722_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) ).

% neg_numeral_le_numeral
tff(fact_6723_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) ).

% not_numeral_le_neg_numeral
tff(fact_6724_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(N))) ).

% not_numeral_le_neg_numeral
tff(fact_6725_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) ).

% not_numeral_le_neg_numeral
tff(fact_6726_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),numeral_numeral_real(N)) ).

% neg_numeral_less_numeral
tff(fact_6727_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),numeral_numeral_rat(N)) ).

% neg_numeral_less_numeral
tff(fact_6728_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) ).

% neg_numeral_less_numeral
tff(fact_6729_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) ).

% not_numeral_less_neg_numeral
tff(fact_6730_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(M)),uminus_uminus_rat(numeral_numeral_rat(N))) ).

% not_numeral_less_neg_numeral
tff(fact_6731_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) ).

% not_numeral_less_neg_numeral
tff(fact_6732_neg__eq__iff__add__eq__0,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( uminus8244633308260627903l_num1(A2) = B2 )
    <=> ( plus_p361126936061061375l_num1(A2,B2) = zero_z3563351764282998399l_num1 ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6733_neg__eq__iff__add__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A2) = B2 )
    <=> ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6734_neg__eq__iff__add__eq__0,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( uminus_uminus_uint32(A2) = B2 )
    <=> ( plus_plus_uint32(A2,B2) = zero_zero_uint32 ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6735_neg__eq__iff__add__eq__0,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(uminus_uminus_real,A2) = B2 )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6736_neg__eq__iff__add__eq__0,axiom,
    ! [A2: rat,B2: rat] :
      ( ( uminus_uminus_rat(A2) = B2 )
    <=> ( plus_plus_rat(A2,B2) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6737_neg__eq__iff__add__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = B2 )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_6738_eq__neg__iff__add__eq__0,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( A2 = uminus8244633308260627903l_num1(B2) )
    <=> ( plus_p361126936061061375l_num1(A2,B2) = zero_z3563351764282998399l_num1 ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6739_eq__neg__iff__add__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 = aa_complex_complex(uminus1482373934393186551omplex,B2) )
    <=> ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6740_eq__neg__iff__add__eq__0,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( A2 = uminus_uminus_uint32(B2) )
    <=> ( plus_plus_uint32(A2,B2) = zero_zero_uint32 ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6741_eq__neg__iff__add__eq__0,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = aa_real_real(uminus_uminus_real,B2) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6742_eq__neg__iff__add__eq__0,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = uminus_uminus_rat(B2) )
    <=> ( plus_plus_rat(A2,B2) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6743_eq__neg__iff__add__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(uminus_uminus_int,B2) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_6744_add_Oinverse__unique,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1(A2,B2) = zero_z3563351764282998399l_num1 )
     => ( uminus8244633308260627903l_num1(A2) = B2 ) ) ).

% add.inverse_unique
tff(fact_6745_add_Oinverse__unique,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A2) = B2 ) ) ).

% add.inverse_unique
tff(fact_6746_add_Oinverse__unique,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( plus_plus_uint32(A2,B2) = zero_zero_uint32 )
     => ( uminus_uminus_uint32(A2) = B2 ) ) ).

% add.inverse_unique
tff(fact_6747_add_Oinverse__unique,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A2) = B2 ) ) ).

% add.inverse_unique
tff(fact_6748_add_Oinverse__unique,axiom,
    ! [A2: rat,B2: rat] :
      ( ( plus_plus_rat(A2,B2) = zero_zero_rat )
     => ( uminus_uminus_rat(A2) = B2 ) ) ).

% add.inverse_unique
tff(fact_6749_add_Oinverse__unique,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A2) = B2 ) ) ).

% add.inverse_unique
tff(fact_6750_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: word_N3645301735248828278l_num1] : plus_p361126936061061375l_num1(uminus8244633308260627903l_num1(A2),A2) = zero_z3563351764282998399l_num1 ).

% ab_group_add_class.ab_left_minus
tff(fact_6751_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),A2) = zero_zero_complex ).

% ab_group_add_class.ab_left_minus
tff(fact_6752_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: uint32] : plus_plus_uint32(uminus_uminus_uint32(A2),A2) = zero_zero_uint32 ).

% ab_group_add_class.ab_left_minus
tff(fact_6753_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),A2) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_6754_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: rat] : plus_plus_rat(uminus_uminus_rat(A2),A2) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_6755_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A2)),A2) = zero_zero_int ).

% ab_group_add_class.ab_left_minus
tff(fact_6756_add__eq__0__iff,axiom,
    ! [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1(A2,B2) = zero_z3563351764282998399l_num1 )
    <=> ( B2 = uminus8244633308260627903l_num1(A2) ) ) ).

% add_eq_0_iff
tff(fact_6757_add__eq__0__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,A2),B2) = zero_zero_complex )
    <=> ( B2 = aa_complex_complex(uminus1482373934393186551omplex,A2) ) ) ).

% add_eq_0_iff
tff(fact_6758_add__eq__0__iff,axiom,
    ! [A2: uint32,B2: uint32] :
      ( ( plus_plus_uint32(A2,B2) = zero_zero_uint32 )
    <=> ( B2 = uminus_uminus_uint32(A2) ) ) ).

% add_eq_0_iff
tff(fact_6759_add__eq__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = zero_zero_real )
    <=> ( B2 = aa_real_real(uminus_uminus_real,A2) ) ) ).

% add_eq_0_iff
tff(fact_6760_add__eq__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( plus_plus_rat(A2,B2) = zero_zero_rat )
    <=> ( B2 = uminus_uminus_rat(A2) ) ) ).

% add_eq_0_iff
tff(fact_6761_add__eq__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2) = zero_zero_int )
    <=> ( B2 = aa_int_int(uminus_uminus_int,A2) ) ) ).

% add_eq_0_iff
tff(fact_6762_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_6763_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_6764_zero__neq__neg__one,axiom,
    zero_zero_rat != uminus_uminus_rat(one_one_rat) ).

% zero_neq_neg_one
tff(fact_6765_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_6766_le__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(4)
tff(fact_6767_le__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),uminus_uminus_rat(one_one_rat)) ).

% le_minus_one_simps(4)
tff(fact_6768_le__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(4)
tff(fact_6769_le__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% le_minus_one_simps(2)
tff(fact_6770_le__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),one_one_rat) ).

% le_minus_one_simps(2)
tff(fact_6771_le__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% le_minus_one_simps(2)
tff(fact_6772_less__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(4)
tff(fact_6773_less__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),uminus_uminus_rat(one_one_rat)) ).

% less_minus_one_simps(4)
tff(fact_6774_less__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(4)
tff(fact_6775_less__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% less_minus_one_simps(2)
tff(fact_6776_less__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),one_one_rat) ).

% less_minus_one_simps(2)
tff(fact_6777_less__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% less_minus_one_simps(2)
tff(fact_6778_numeral__neq__neg__one,axiom,
    ! [N: num] : numera6690914467698888265omplex(N) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_6779_numeral__neq__neg__one,axiom,
    ! [N: num] : numeral_numeral_real(N) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_6780_numeral__neq__neg__one,axiom,
    ! [N: num] : numeral_numeral_rat(N) != uminus_uminus_rat(one_one_rat) ).

% numeral_neq_neg_one
tff(fact_6781_numeral__neq__neg__one,axiom,
    ! [N: num] : aa_num_int(numeral_numeral_int,N) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_6782_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(N)) ).

% one_neq_neg_numeral
tff(fact_6783_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_real != aa_real_real(uminus_uminus_real,numeral_numeral_real(N)) ).

% one_neq_neg_numeral
tff(fact_6784_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_rat != uminus_uminus_rat(numeral_numeral_rat(N)) ).

% one_neq_neg_numeral
tff(fact_6785_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)) ).

% one_neq_neg_numeral
tff(fact_6786_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(W)),uminus8244633308260627903l_num1(Xb)) = aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(Xb),uminus8244633308260627903l_num1(numera7442385471795722001l_num1(W))) ).

% numeral_times_minus_swap
tff(fact_6787_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(W)),aa_complex_complex(uminus1482373934393186551omplex,Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% numeral_times_minus_swap
tff(fact_6788_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: uint32] : aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(W)),uminus_uminus_uint32(Xb)) = aa_uint32_uint32(times_times_uint32(Xb),uminus_uminus_uint32(numera9087168376688890119uint32(W))) ).

% numeral_times_minus_swap
tff(fact_6789_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: real] : aa_real_real(times_times_real(numeral_numeral_real(W)),aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(times_times_real(Xb),aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) ).

% numeral_times_minus_swap
tff(fact_6790_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: rat] : aa_rat_rat(times_times_rat(numeral_numeral_rat(W)),uminus_uminus_rat(Xb)) = aa_rat_rat(times_times_rat(Xb),uminus_uminus_rat(numeral_numeral_rat(W))) ).

% numeral_times_minus_swap
tff(fact_6791_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),aa_int_int(uminus_uminus_int,Xb)) = aa_int_int(aa_int_fun_int_int(times_times_int,Xb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).

% numeral_times_minus_swap
tff(fact_6792_nonzero__minus__divide__right,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),aa_complex_complex(uminus1482373934393186551omplex,B2)) ) ) ).

% nonzero_minus_divide_right
tff(fact_6793_nonzero__minus__divide__right,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A2),B2)) = aa_real_real(divide_divide_real(A2),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% nonzero_minus_divide_right
tff(fact_6794_nonzero__minus__divide__right,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( uminus_uminus_rat(divide_divide_rat(A2,B2)) = divide_divide_rat(A2,uminus_uminus_rat(B2)) ) ) ).

% nonzero_minus_divide_right
tff(fact_6795_nonzero__minus__divide__divide,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2) ) ) ).

% nonzero_minus_divide_divide
tff(fact_6796_nonzero__minus__divide__divide,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A2)),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(divide_divide_real(A2),B2) ) ) ).

% nonzero_minus_divide_divide
tff(fact_6797_nonzero__minus__divide__divide,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( divide_divide_rat(uminus_uminus_rat(A2),uminus_uminus_rat(B2)) = divide_divide_rat(A2,B2) ) ) ).

% nonzero_minus_divide_divide
tff(fact_6798_square__eq__1__iff,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Xb),Xb) = one_one_complex )
    <=> ( ( Xb = one_one_complex )
        | ( Xb = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_6799_square__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(times_times_real(Xb),Xb) = one_one_real )
    <=> ( ( Xb = one_one_real )
        | ( Xb = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_6800_square__eq__1__iff,axiom,
    ! [Xb: rat] :
      ( ( aa_rat_rat(times_times_rat(Xb),Xb) = one_one_rat )
    <=> ( ( Xb = one_one_rat )
        | ( Xb = uminus_uminus_rat(one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_6801_square__eq__1__iff,axiom,
    ! [Xb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb) = one_one_int )
    <=> ( ( Xb = one_one_int )
        | ( Xb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_6802_dvd__div__neg,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)) ) ) ).

% dvd_div_neg
tff(fact_6803_dvd__div__neg,axiom,
    ! [B2: complex,A2: complex] :
      ( aa_complex_o(dvd_dvd_complex(B2),A2)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2)) ) ) ).

% dvd_div_neg
tff(fact_6804_dvd__div__neg,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A2)
     => ( aa_real_real(divide_divide_real(A2),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A2),B2)) ) ) ).

% dvd_div_neg
tff(fact_6805_dvd__div__neg,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A2)
     => ( divide_divide_rat(A2,uminus_uminus_rat(B2)) = uminus_uminus_rat(divide_divide_rat(A2,B2)) ) ) ).

% dvd_div_neg
tff(fact_6806_dvd__div__neg,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% dvd_div_neg
tff(fact_6807_dvd__neg__div,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2)
     => ( aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),B2)) ) ) ).

% dvd_neg_div
tff(fact_6808_dvd__neg__div,axiom,
    ! [B2: complex,A2: complex] :
      ( aa_complex_o(dvd_dvd_complex(B2),A2)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(uminus1482373934393186551omplex,A2)),B2) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2)) ) ) ).

% dvd_neg_div
tff(fact_6809_dvd__neg__div,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A2)
     => ( aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A2)),B2) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A2),B2)) ) ) ).

% dvd_neg_div
tff(fact_6810_dvd__neg__div,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A2)
     => ( divide_divide_rat(uminus_uminus_rat(A2),B2) = uminus_uminus_rat(divide_divide_rat(A2,B2)) ) ) ).

% dvd_neg_div
tff(fact_6811_dvd__neg__div,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A2)),B2) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% dvd_neg_div
tff(fact_6812_Heap_Osize_I2_J,axiom,
    ! [Xb: fun_he1353482274668092617it_nat] : size_s6287829766004316056on_nat(heap_T5286843759275942675on_nat(Xb)) = aa_nat_nat(suc,zero_zero_nat) ).

% Heap.size(2)
tff(fact_6813_Heap_Osize_I2_J,axiom,
    ! [Xb: fun_he1183298388918465219it_nat] : size_s2700093152935483318Heap_o(heap_Time_Heap_o2(Xb)) = aa_nat_nat(suc,zero_zero_nat) ).

% Heap.size(2)
tff(fact_6814_Heap_Osize_I2_J,axiom,
    ! [Xb: fun_he3045699446853741534it_nat] : size_s8425857057747876397_VEBTi(heap_T1489671670754571048_VEBTi(Xb)) = aa_nat_nat(suc,zero_zero_nat) ).

% Heap.size(2)
tff(fact_6815_neg__numeral__le__zero,axiom,
    ! [N: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_6816_neg__numeral__le__zero,axiom,
    ! [N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(N))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_6817_neg__numeral__le__zero,axiom,
    ! [N: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_6818_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) ).

% not_zero_le_neg_numeral
tff(fact_6819_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(N))) ).

% not_zero_le_neg_numeral
tff(fact_6820_not__zero__le__neg__numeral,axiom,
    ! [N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) ).

% not_zero_le_neg_numeral
tff(fact_6821_neg__numeral__less__zero,axiom,
    ! [N: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_6822_neg__numeral__less__zero,axiom,
    ! [N: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(N))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_6823_neg__numeral__less__zero,axiom,
    ! [N: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_6824_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) ).

% not_zero_less_neg_numeral
tff(fact_6825_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(N))) ).

% not_zero_less_neg_numeral
tff(fact_6826_not__zero__less__neg__numeral,axiom,
    ! [N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) ).

% not_zero_less_neg_numeral
tff(fact_6827_le__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(3)
tff(fact_6828_le__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(one_one_rat)) ).

% le_minus_one_simps(3)
tff(fact_6829_le__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(3)
tff(fact_6830_le__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% le_minus_one_simps(1)
tff(fact_6831_le__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),zero_zero_rat) ).

% le_minus_one_simps(1)
tff(fact_6832_le__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% le_minus_one_simps(1)
tff(fact_6833_less__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(3)
tff(fact_6834_less__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),uminus_uminus_rat(one_one_rat)) ).

% less_minus_one_simps(3)
tff(fact_6835_less__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(3)
tff(fact_6836_less__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% less_minus_one_simps(1)
tff(fact_6837_less__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),zero_zero_rat) ).

% less_minus_one_simps(1)
tff(fact_6838_less__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% less_minus_one_simps(1)
tff(fact_6839_neg__numeral__le__one,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real) ).

% neg_numeral_le_one
tff(fact_6840_neg__numeral__le__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_6841_neg__numeral__le__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) ).

% neg_numeral_le_one
tff(fact_6842_neg__one__le__numeral,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),numeral_numeral_real(M)) ).

% neg_one_le_numeral
tff(fact_6843_neg__one__le__numeral,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(M)) ).

% neg_one_le_numeral
tff(fact_6844_neg__one__le__numeral,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M)) ).

% neg_one_le_numeral
tff(fact_6845_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_6846_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(M))),uminus_uminus_rat(one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_6847_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_6848_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_6849_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,numeral_numeral_rat(M)),uminus_uminus_rat(one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_6850_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_6851_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(M))) ).

% not_one_le_neg_numeral
tff(fact_6852_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),uminus_uminus_rat(numeral_numeral_rat(M))) ).

% not_one_le_neg_numeral
tff(fact_6853_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_one_le_neg_numeral
tff(fact_6854_neg__numeral__less__one,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(M))),one_one_real) ).

% neg_numeral_less_one
tff(fact_6855_neg__numeral__less__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(M))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_6856_neg__numeral__less__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) ).

% neg_numeral_less_one
tff(fact_6857_neg__one__less__numeral,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),numeral_numeral_real(M)) ).

% neg_one_less_numeral
tff(fact_6858_neg__one__less__numeral,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(M)) ).

% neg_one_less_numeral
tff(fact_6859_neg__one__less__numeral,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M)) ).

% neg_one_less_numeral
tff(fact_6860_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,numeral_numeral_real(M)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_less_neg_one
tff(fact_6861_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,numeral_numeral_rat(M)),uminus_uminus_rat(one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_6862_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_6863_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(M))) ).

% not_one_less_neg_numeral
tff(fact_6864_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),uminus_uminus_rat(numeral_numeral_rat(M))) ).

% not_one_less_neg_numeral
tff(fact_6865_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_one_less_neg_numeral
tff(fact_6866_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,numeral_numeral_real(M))) ).

% not_neg_one_less_neg_numeral
tff(fact_6867_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(M))) ).

% not_neg_one_less_neg_numeral
tff(fact_6868_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_6869_uminus__numeral__One,axiom,
    uminus8244633308260627903l_num1(numera7442385471795722001l_num1(one)) = uminus8244633308260627903l_num1(one_on7727431528512463931l_num1) ).

% uminus_numeral_One
tff(fact_6870_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_6871_uminus__numeral__One,axiom,
    uminus_uminus_uint32(numera9087168376688890119uint32(one)) = uminus_uminus_uint32(one_one_uint32) ).

% uminus_numeral_One
tff(fact_6872_uminus__numeral__One,axiom,
    aa_real_real(uminus_uminus_real,numeral_numeral_real(one)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% uminus_numeral_One
tff(fact_6873_uminus__numeral__One,axiom,
    uminus_uminus_rat(numeral_numeral_rat(one)) = uminus_uminus_rat(one_one_rat) ).

% uminus_numeral_One
tff(fact_6874_uminus__numeral__One,axiom,
    aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% uminus_numeral_One
tff(fact_6875_mult__1s__ring__1_I2_J,axiom,
    ! [B2: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(B2),uminus8244633308260627903l_num1(numera7442385471795722001l_num1(one))) = uminus8244633308260627903l_num1(B2) ).

% mult_1s_ring_1(2)
tff(fact_6876_mult__1s__ring__1_I2_J,axiom,
    ! [B2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))) = aa_complex_complex(uminus1482373934393186551omplex,B2) ).

% mult_1s_ring_1(2)
tff(fact_6877_mult__1s__ring__1_I2_J,axiom,
    ! [B2: uint32] : aa_uint32_uint32(times_times_uint32(B2),uminus_uminus_uint32(numera9087168376688890119uint32(one))) = uminus_uminus_uint32(B2) ).

% mult_1s_ring_1(2)
tff(fact_6878_mult__1s__ring__1_I2_J,axiom,
    ! [B2: real] : aa_real_real(times_times_real(B2),aa_real_real(uminus_uminus_real,numeral_numeral_real(one))) = aa_real_real(uminus_uminus_real,B2) ).

% mult_1s_ring_1(2)
tff(fact_6879_mult__1s__ring__1_I2_J,axiom,
    ! [B2: rat] : aa_rat_rat(times_times_rat(B2),uminus_uminus_rat(numeral_numeral_rat(one))) = uminus_uminus_rat(B2) ).

% mult_1s_ring_1(2)
tff(fact_6880_mult__1s__ring__1_I2_J,axiom,
    ! [B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) = aa_int_int(uminus_uminus_int,B2) ).

% mult_1s_ring_1(2)
tff(fact_6881_mult__1s__ring__1_I1_J,axiom,
    ! [B2: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(one))),B2) = uminus8244633308260627903l_num1(B2) ).

% mult_1s_ring_1(1)
tff(fact_6882_mult__1s__ring__1_I1_J,axiom,
    ! [B2: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))),B2) = aa_complex_complex(uminus1482373934393186551omplex,B2) ).

% mult_1s_ring_1(1)
tff(fact_6883_mult__1s__ring__1_I1_J,axiom,
    ! [B2: uint32] : aa_uint32_uint32(times_times_uint32(uminus_uminus_uint32(numera9087168376688890119uint32(one))),B2) = uminus_uminus_uint32(B2) ).

% mult_1s_ring_1(1)
tff(fact_6884_mult__1s__ring__1_I1_J,axiom,
    ! [B2: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(one))),B2) = aa_real_real(uminus_uminus_real,B2) ).

% mult_1s_ring_1(1)
tff(fact_6885_mult__1s__ring__1_I1_J,axiom,
    ! [B2: rat] : aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(one))),B2) = uminus_uminus_rat(B2) ).

% mult_1s_ring_1(1)
tff(fact_6886_mult__1s__ring__1_I1_J,axiom,
    ! [B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))),B2) = aa_int_int(uminus_uminus_int,B2) ).

% mult_1s_ring_1(1)
tff(fact_6887_divide__eq__minus__1__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( B2 != zero_zero_complex )
        & ( A2 = aa_complex_complex(uminus1482373934393186551omplex,B2) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_6888_divide__eq__minus__1__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(divide_divide_real(A2),B2) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B2 != zero_zero_real )
        & ( A2 = aa_real_real(uminus_uminus_real,B2) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_6889_divide__eq__minus__1__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( divide_divide_rat(A2,B2) = uminus_uminus_rat(one_one_rat) )
    <=> ( ( B2 != zero_zero_rat )
        & ( A2 = uminus_uminus_rat(B2) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_6890_eq__minus__divide__eq,axiom,
    ! [A2: complex,B2: complex,C: complex] :
      ( ( A2 = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C)) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),C) = aa_complex_complex(uminus1482373934393186551omplex,B2),A2 = zero_zero_complex) ) ).

% eq_minus_divide_eq
tff(fact_6891_eq__minus__divide__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( ( A2 = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C)) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(A2),C) = aa_real_real(uminus_uminus_real,B2),A2 = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_6892_eq__minus__divide__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( ( A2 = uminus_uminus_rat(divide_divide_rat(B2,C)) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A2),C) = uminus_uminus_rat(B2),A2 = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_6893_minus__divide__eq__eq,axiom,
    ! [B2: complex,C: complex,A2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C)) = A2 )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B2) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,A2),C),A2 = zero_zero_complex) ) ).

% minus_divide_eq_eq
tff(fact_6894_minus__divide__eq__eq,axiom,
    ! [B2: real,C: real,A2: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C)) = A2 )
    <=> $ite(C != zero_zero_real,aa_real_real(uminus_uminus_real,B2) = aa_real_real(times_times_real(A2),C),A2 = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_6895_minus__divide__eq__eq,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( ( uminus_uminus_rat(divide_divide_rat(B2,C)) = A2 )
    <=> $ite(C != zero_zero_rat,uminus_uminus_rat(B2) = aa_rat_rat(times_times_rat(A2),C),A2 = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_6896_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: complex,A2: complex,C: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2)) = C )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B2) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6897_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: real,A2: real,C: real] :
      ( ( B2 != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A2),B2)) = C )
      <=> ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(times_times_real(C),B2) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6898_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: rat,A2: rat,C: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( uminus_uminus_rat(divide_divide_rat(A2,B2)) = C )
      <=> ( uminus_uminus_rat(A2) = aa_rat_rat(times_times_rat(C),B2) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_6899_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: complex,C: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( C = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),B2)) )
      <=> ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),B2) = aa_complex_complex(uminus1482373934393186551omplex,A2) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6900_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: real,C: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( C = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A2),B2)) )
      <=> ( aa_real_real(times_times_real(C),B2) = aa_real_real(uminus_uminus_real,A2) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6901_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( C = uminus_uminus_rat(divide_divide_rat(A2,B2)) )
      <=> ( aa_rat_rat(times_times_rat(C),B2) = uminus_uminus_rat(A2) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_6902_power__minus,axiom,
    ! [A2: code_integer,N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ).

% power_minus
tff(fact_6903_power__minus,axiom,
    ! [A2: complex,N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),N) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N)) ).

% power_minus
tff(fact_6904_power__minus,axiom,
    ! [A2: uint32,N: nat] : aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(A2)),N) = aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(one_one_uint32)),N)),aa_nat_uint32(power_power_uint32(A2),N)) ).

% power_minus
tff(fact_6905_power__minus,axiom,
    ! [A2: real,N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),N) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ).

% power_minus
tff(fact_6906_power__minus,axiom,
    ! [A2: rat,N: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(A2)),N) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N)),aa_nat_rat(power_power_rat(A2),N)) ).

% power_minus
tff(fact_6907_power__minus,axiom,
    ! [A2: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N)),aa_nat_int(power_power_int(A2),N)) ).

% power_minus
tff(fact_6908_power__minus__Bit0,axiom,
    ! [Xb: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6909_power__minus__Bit0,axiom,
    ! [Xb: complex,K: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6910_power__minus__Bit0,axiom,
    ! [Xb: uint32,K: num] : aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_uint32(power_power_uint32(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6911_power__minus__Bit0,axiom,
    ! [Xb: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6912_power__minus__Bit0,axiom,
    ! [Xb: rat,K: num] : aa_nat_rat(power_power_rat(uminus_uminus_rat(Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6913_power__minus__Bit0,axiom,
    ! [Xb: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_6914_power__minus__Bit1,axiom,
    ! [Xb: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6915_power__minus__Bit1,axiom,
    ! [Xb: complex,K: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6916_power__minus__Bit1,axiom,
    ! [Xb: uint32,K: num] : aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = uminus_uminus_uint32(aa_nat_uint32(power_power_uint32(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6917_power__minus__Bit1,axiom,
    ! [Xb: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6918_power__minus__Bit1,axiom,
    ! [Xb: rat,K: num] : aa_nat_rat(power_power_rat(uminus_uminus_rat(Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = uminus_uminus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6919_power__minus__Bit1,axiom,
    ! [Xb: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_6920_real__add__less__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(uminus_uminus_real,Xb)) ) ).

% real_add_less_0_iff
tff(fact_6921_real__0__less__add__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Xb)),Y) ) ).

% real_0_less_add_iff
tff(fact_6922_real__0__le__add__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xb)),Y) ) ).

% real_0_le_add_iff
tff(fact_6923_real__add__le__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,Xb)) ) ).

% real_add_le_0_iff
tff(fact_6924_tanh__real__gt__neg1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,Xb)) ).

% tanh_real_gt_neg1
tff(fact_6925_less__minus__divide__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),aa_real_real(uminus_uminus_real,B2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(times_times_real(A2),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)) ) ) ).

% less_minus_divide_eq
tff(fact_6926_less__minus__divide__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),uminus_uminus_rat(divide_divide_rat(B2,C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),uminus_uminus_rat(B2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B2)),aa_rat_rat(times_times_rat(A2),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_6927_minus__divide__less__eq,axiom,
    ! [B2: real,C: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))),A2)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(times_times_real(A2),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),aa_real_real(uminus_uminus_real,B2)),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)) ) ) ).

% minus_divide_less_eq
tff(fact_6928_minus__divide__less__eq,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B2,C))),A2)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B2)),aa_rat_rat(times_times_rat(A2),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),uminus_uminus_rat(B2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)) ) ) ).

% minus_divide_less_eq
tff(fact_6929_neg__less__minus__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(times_times_real(A2),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_6930_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),uminus_uminus_rat(divide_divide_rat(B2,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B2)),aa_rat_rat(times_times_rat(A2),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_6931_neg__minus__divide__less__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))),A2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_6932_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B2,C))),A2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),uminus_uminus_rat(B2)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_6933_pos__less__minus__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A2),C)),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_6934_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),uminus_uminus_rat(divide_divide_rat(B2,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C)),uminus_uminus_rat(B2)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_6935_pos__minus__divide__less__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))),A2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(times_times_real(A2),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_6936_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(divide_divide_rat(B2,C))),A2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(B2)),aa_rat_rat(times_times_rat(A2),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_6937_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: complex,C: complex,W: num] :
      ( ( aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> $ite(C != zero_zero_complex,B2 = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6938_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( ( aa_real_real(divide_divide_real(B2),C) = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) )
    <=> $ite(C != zero_zero_real,B2 = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6939_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( ( divide_divide_rat(B2,C) = uminus_uminus_rat(numeral_numeral_rat(W)) )
    <=> $ite(C != zero_zero_rat,B2 = aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C),uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_6940_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,B2),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) = B2,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6941_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = aa_real_real(divide_divide_real(B2),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C) = B2,aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6942_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(W)) = divide_divide_rat(B2,C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C) = B2,uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_6943_add__divide__eq__if__simps_I3_J,axiom,
    ! [A2: complex,Z: complex,B2: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),Z))),B2) = $ite(Z = zero_zero_complex,B2,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B2),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6944_add__divide__eq__if__simps_I3_J,axiom,
    ! [A2: real,Z: real,B2: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A2),Z))),B2) = $ite(Z = zero_zero_real,B2,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(times_times_real(B2),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6945_add__divide__eq__if__simps_I3_J,axiom,
    ! [A2: rat,Z: rat,B2: rat] :
      plus_plus_rat(uminus_uminus_rat(divide_divide_rat(A2,Z)),B2) = $ite(Z = zero_zero_rat,B2,divide_divide_rat(plus_plus_rat(uminus_uminus_rat(A2),aa_rat_rat(times_times_rat(B2),Z)),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_6946_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Z))),Y) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6947_minus__divide__add__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Xb),Z))),Y) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(times_times_real(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6948_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( plus_plus_rat(uminus_uminus_rat(divide_divide_rat(Xb,Z)),Y) = divide_divide_rat(plus_plus_rat(uminus_uminus_rat(Xb),aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_6949_add__divide__eq__if__simps_I6_J,axiom,
    ! [A2: complex,Z: complex,B2: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),Z))),B2) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B2),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B2),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6950_add__divide__eq__if__simps_I6_J,axiom,
    ! [A2: real,Z: real,B2: real] :
      aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A2),Z))),B2) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B2),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,A2)),aa_real_real(times_times_real(B2),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6951_add__divide__eq__if__simps_I6_J,axiom,
    ! [A2: rat,Z: rat,B2: rat] :
      minus_minus_rat(uminus_uminus_rat(divide_divide_rat(A2,Z)),B2) = $ite(Z = zero_zero_rat,uminus_uminus_rat(B2),divide_divide_rat(minus_minus_rat(uminus_uminus_rat(A2),aa_rat_rat(times_times_rat(B2),Z)),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_6952_add__divide__eq__if__simps_I5_J,axiom,
    ! [A2: complex,Z: complex,B2: complex] :
      aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,A2),Z)),B2) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B2),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,A2),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,B2),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6953_add__divide__eq__if__simps_I5_J,axiom,
    ! [A2: real,Z: real,B2: real] :
      aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A2),Z)),B2) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B2),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(A2),aa_real_real(times_times_real(B2),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6954_add__divide__eq__if__simps_I5_J,axiom,
    ! [A2: rat,Z: rat,B2: rat] :
      minus_minus_rat(divide_divide_rat(A2,Z),B2) = $ite(Z = zero_zero_rat,uminus_uminus_rat(B2),divide_divide_rat(minus_minus_rat(A2,aa_rat_rat(times_times_rat(B2),Z)),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_6955_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,Xb),Z))),Y) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6956_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Xb),Z))),Y) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,Xb)),aa_real_real(times_times_real(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6957_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( minus_minus_rat(uminus_uminus_rat(divide_divide_rat(Xb,Z)),Y) = divide_divide_rat(minus_minus_rat(uminus_uminus_rat(Xb),aa_rat_rat(times_times_rat(Y),Z)),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_6958_even__minus,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2) ) ).

% even_minus
tff(fact_6959_even__minus,axiom,
    ! [A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),uminus8244633308260627903l_num1(A2))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2) ) ).

% even_minus
tff(fact_6960_even__minus,axiom,
    ! [A2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),uminus_uminus_uint32(A2))
    <=> aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2) ) ).

% even_minus
tff(fact_6961_even__minus,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2) ) ).

% even_minus
tff(fact_6962_power2__eq__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6963_power2__eq__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_complex_complex(uminus1482373934393186551omplex,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6964_power2__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6965_power2__eq__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y )
        | ( Xb = uminus_uminus_rat(Y) ) ) ) ).

% power2_eq_iff
tff(fact_6966_power2__eq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% power2_eq_iff
tff(fact_6967_pred__equals__eq2,axiom,
    ! [R4: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ! [X: num,Xa3: num] :
          ( member7279096912039735102um_num(aa_num3156681643174609981um_num(product_Pair_num_num(X),Xa3),R4)
        <=> member7279096912039735102um_num(aa_num3156681643174609981um_num(product_Pair_num_num(X),Xa3),S3) )
    <=> ( R4 = S3 ) ) ).

% pred_equals_eq2
tff(fact_6968_pred__equals__eq2,axiom,
    ! [R4: set_Pr563407847431865468T_VEBT,S3: set_Pr563407847431865468T_VEBT] :
      ( ! [X: nat,Xa3: produc4813437837504472865T_VEBT] :
          ( member306291179834725981T_VEBT(aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(X),Xa3),R4)
        <=> member306291179834725981T_VEBT(aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(X),Xa3),S3) )
    <=> ( R4 = S3 ) ) ).

% pred_equals_eq2
tff(fact_6969_pred__equals__eq2,axiom,
    ! [R4: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ! [X: nat,Xa3: num] :
          ( member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(X),Xa3),R4)
        <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(X),Xa3),S3) )
    <=> ( R4 = S3 ) ) ).

% pred_equals_eq2
tff(fact_6970_pred__equals__eq2,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Xa3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa3),R4)
        <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa3),S3) )
    <=> ( R4 = S3 ) ) ).

% pred_equals_eq2
tff(fact_6971_pred__equals__eq2,axiom,
    ! [R4: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ! [X: int,Xa3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa3),R4)
        <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa3),S3) )
    <=> ( R4 = S3 ) ) ).

% pred_equals_eq2
tff(fact_6972_le__minus__divide__eq,axiom,
    ! [A2: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),C)),aa_real_real(uminus_uminus_real,B2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(times_times_real(A2),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)) ) ) ).

% le_minus_divide_eq
tff(fact_6973_le__minus__divide__eq,axiom,
    ! [A2: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),uminus_uminus_rat(divide_divide_rat(B2,C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C)),uminus_uminus_rat(B2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B2)),aa_rat_rat(times_times_rat(A2),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_6974_minus__divide__le__eq,axiom,
    ! [B2: real,C: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))),A2)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(times_times_real(A2),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),C)),aa_real_real(uminus_uminus_real,B2)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)) ) ) ).

% minus_divide_le_eq
tff(fact_6975_minus__divide__le__eq,axiom,
    ! [B2: rat,C: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B2,C))),A2)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B2)),aa_rat_rat(times_times_rat(A2),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C)),uminus_uminus_rat(B2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)) ) ) ).

% minus_divide_le_eq
tff(fact_6976_neg__le__minus__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(times_times_real(A2),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_6977_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),uminus_uminus_rat(divide_divide_rat(B2,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B2)),aa_rat_rat(times_times_rat(A2),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_6978_neg__minus__divide__le__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))),A2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),C)),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_6979_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B2,C))),A2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C)),uminus_uminus_rat(B2)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_6980_pos__le__minus__divide__eq,axiom,
    ! [C: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A2),C)),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_6981_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),uminus_uminus_rat(divide_divide_rat(B2,C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C)),uminus_uminus_rat(B2)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_6982_pos__minus__divide__le__eq,axiom,
    ! [C: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B2),C))),A2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(times_times_real(A2),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_6983_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(divide_divide_rat(B2,C))),A2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(B2)),aa_rat_rat(times_times_rat(A2),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_6984_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B2),C)),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B2),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_6985_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,divide_divide_rat(B2,C)),uminus_uminus_rat(numeral_numeral_rat(W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_6986_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),aa_real_real(divide_divide_real(B2),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B2),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_6987_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(W))),divide_divide_rat(B2,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B2),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(numeral_numeral_rat(W))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_6988_power2__eq__1__iff,axiom,
    ! [A2: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( ( A2 = one_one_Code_integer )
        | ( A2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_6989_power2__eq__1__iff,axiom,
    ! [A2: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
    <=> ( ( A2 = one_one_complex )
        | ( A2 = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% power2_eq_1_iff
tff(fact_6990_power2__eq__1__iff,axiom,
    ! [A2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( ( A2 = one_one_real )
        | ( A2 = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% power2_eq_1_iff
tff(fact_6991_power2__eq__1__iff,axiom,
    ! [A2: rat] :
      ( ( aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( ( A2 = one_one_rat )
        | ( A2 = uminus_uminus_rat(one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_6992_power2__eq__1__iff,axiom,
    ! [A2: int] :
      ( ( aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( ( A2 = one_one_int )
        | ( A2 = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% power2_eq_1_iff
tff(fact_6993_uminus__power__if,axiom,
    ! [A2: code_integer,N: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_Code_integer(power_8256067586552552935nteger(A2),N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N))) ).

% uminus_power_if
tff(fact_6994_uminus__power__if,axiom,
    ! [A2: complex,N: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N),aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),N))) ).

% uminus_power_if
tff(fact_6995_uminus__power__if,axiom,
    ! [A2: uint32,N: nat] :
      aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(A2)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_uint32(power_power_uint32(A2),N),uminus_uminus_uint32(aa_nat_uint32(power_power_uint32(A2),N))) ).

% uminus_power_if
tff(fact_6996_uminus__power__if,axiom,
    ! [A2: real,N: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N),aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N))) ).

% uminus_power_if
tff(fact_6997_uminus__power__if,axiom,
    ! [A2: rat,N: nat] :
      aa_nat_rat(power_power_rat(uminus_uminus_rat(A2)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_rat(power_power_rat(A2),N),uminus_uminus_rat(aa_nat_rat(power_power_rat(A2),N))) ).

% uminus_power_if
tff(fact_6998_uminus__power__if,axiom,
    ! [A2: int,N: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_int(power_power_int(A2),N),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A2),N))) ).

% uminus_power_if
tff(fact_6999_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7000_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7001_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(one_one_uint32)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(one_one_uint32)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7002_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7003_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7004_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7005_realpow__square__minus__le,axiom,
    ! [U: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% realpow_square_minus_le
tff(fact_7006_ln__add__one__self__le__self2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb) ) ).

% ln_add_one_self_le_self2
tff(fact_7007_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B2),C)),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B2),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,numeral_numeral_real(W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_7008_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(B2,C)),uminus_uminus_rat(numeral_numeral_rat(W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_7009_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),aa_real_real(divide_divide_real(B2),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),B2),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_7010_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(W))),divide_divide_rat(B2,C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),B2),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(uminus_uminus_rat(numeral_numeral_rat(W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(numeral_numeral_rat(W))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_7011_square__le__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Xb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_7012_square__le__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ) ) ).

% square_le_1
tff(fact_7013_square__le__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(one_one_rat)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_7014_square__le__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_7015_minus__power__mult__self,axiom,
    ! [A2: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),N)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_7016_minus__power__mult__self,axiom,
    ! [A2: complex,N: nat] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),N)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)),N)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_7017_minus__power__mult__self,axiom,
    ! [A2: uint32,N: nat] : aa_uint32_uint32(times_times_uint32(aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(A2)),N)),aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(A2)),N)) = aa_nat_uint32(power_power_uint32(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_7018_minus__power__mult__self,axiom,
    ! [A2: real,N: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_7019_minus__power__mult__self,axiom,
    ! [A2: rat,N: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(A2)),N)),aa_nat_rat(power_power_rat(uminus_uminus_rat(A2)),N)) = aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_7020_minus__power__mult__self,axiom,
    ! [A2: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),N)) = aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% minus_power_mult_self
tff(fact_7021_minus__one__power__iff,axiom,
    ! [N: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% minus_one_power_iff
tff(fact_7022_minus__one__power__iff,axiom,
    ! [N: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),one_one_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) ).

% minus_one_power_iff
tff(fact_7023_minus__one__power__iff,axiom,
    ! [N: nat] :
      aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(one_one_uint32)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),one_one_uint32,uminus_uminus_uint32(one_one_uint32)) ).

% minus_one_power_iff
tff(fact_7024_minus__one__power__iff,axiom,
    ! [N: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% minus_one_power_iff
tff(fact_7025_minus__one__power__iff,axiom,
    ! [N: nat] :
      aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),one_one_rat,uminus_uminus_rat(one_one_rat)) ).

% minus_one_power_iff
tff(fact_7026_minus__one__power__iff,axiom,
    ! [N: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% minus_one_power_iff
tff(fact_7027_ln__one__minus__pos__upper__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(minus_minus_real(one_one_real),Xb))),aa_real_real(uminus_uminus_real,Xb)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_7028_vebt__memberi_Omono,axiom,
    ! [Xb: produc3881548065746020326Ti_nat] : comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aTP_Lamm_sy(Xb)) ).

% vebt_memberi.mono
tff(fact_7029_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_7030_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_7031_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_uint32(power_power_uint32(uminus_uminus_uint32(one_one_uint32)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = uminus_uminus_uint32(one_one_uint32) ).

% power_minus1_odd
tff(fact_7032_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_7033_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_rat(power_power_rat(uminus_uminus_rat(one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = uminus_uminus_rat(one_one_rat) ).

% power_minus1_odd
tff(fact_7034_power__minus1__odd,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_7035_subrelI,axiom,
    ! [R: set_Pr8218934625190621173um_num,S: set_Pr8218934625190621173um_num] :
      ( ! [X2: num,Y2: num] :
          ( member7279096912039735102um_num(aa_num3156681643174609981um_num(product_Pair_num_num(X2),Y2),R)
         => member7279096912039735102um_num(aa_num3156681643174609981um_num(product_Pair_num_num(X2),Y2),S) )
     => ord_le880128212290418581um_num(R,S) ) ).

% subrelI
tff(fact_7036_subrelI,axiom,
    ! [R: set_Pr563407847431865468T_VEBT,S: set_Pr563407847431865468T_VEBT] :
      ( ! [X2: nat,Y2: produc4813437837504472865T_VEBT] :
          ( member306291179834725981T_VEBT(aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(X2),Y2),R)
         => member306291179834725981T_VEBT(aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(X2),Y2),S) )
     => ord_le6438908469242860764T_VEBT(R,S) ) ).

% subrelI
tff(fact_7037_subrelI,axiom,
    ! [R: set_Pr6200539531224447659at_num,S: set_Pr6200539531224447659at_num] :
      ( ! [X2: nat,Y2: num] :
          ( member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(X2),Y2),R)
         => member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(X2),Y2),S) )
     => ord_le8085105155179020875at_num(R,S) ) ).

% subrelI
tff(fact_7038_subrelI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X2: nat,Y2: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y2),R)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X2),Y2),S) )
     => ord_le3146513528884898305at_nat(R,S) ) ).

% subrelI
tff(fact_7039_subrelI,axiom,
    ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X2: int,Y2: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2),R)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X2),Y2),S) )
     => ord_le2843351958646193337nt_int(R,S) ) ).

% subrelI
tff(fact_7040_VEBT__internal_Ovebt__memberi_H_Omono,axiom,
    ! [Xb: produc3960855945107176009Ti_nat] : comple5241396456474302457Heap_o(partia3290229181235258227Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aTP_Lamm_to(Xb)) ).

% VEBT_internal.vebt_memberi'.mono
tff(fact_7041_of__int__code__if,axiom,
    ! [K: int] :
      ring_17408606157368542149l_num1(K) = $ite(
        K = zero_zero_int,
        zero_z3563351764282998399l_num1,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          uminus8244633308260627903l_num1(ring_17408606157368542149l_num1(aa_int_int(uminus_uminus_int,K))),
          $let(
            l: word_N3645301735248828278l_num1,
            l:= aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),ring_17408606157368542149l_num1(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p361126936061061375l_num1(l,one_on7727431528512463931l_num1)) ) ) ) ).

% of_int_code_if
tff(fact_7042_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_complex(ring_17405671764205052669omplex,K) = $ite(
        K = zero_zero_int,
        zero_zero_complex,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: complex,
            l:= aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,l),one_one_complex)) ) ) ) ).

% of_int_code_if
tff(fact_7043_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_uint32(ring_1_of_int_uint32,K) = $ite(
        K = zero_zero_int,
        zero_zero_uint32,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          uminus_uminus_uint32(aa_int_uint32(ring_1_of_int_uint32,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: uint32,
            l:= aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_int_uint32(ring_1_of_int_uint32,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_plus_uint32(l,one_one_uint32)) ) ) ) ).

% of_int_code_if
tff(fact_7044_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_real(ring_1_of_int_real,K) = $ite(
        K = zero_zero_int,
        zero_zero_real,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: real,
            l:= aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,l),one_one_real)) ) ) ) ).

% of_int_code_if
tff(fact_7045_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_rat(ring_1_of_int_rat,K) = $ite(
        K = zero_zero_int,
        zero_zero_rat,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          uminus_uminus_rat(aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: rat,
            l:= aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_plus_rat(l,one_one_rat)) ) ) ) ).

% of_int_code_if
tff(fact_7046_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_int(ring_1_of_int_int,K) = $ite(
        K = zero_zero_int,
        zero_zero_int,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: int,
            l:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_int_int(aa_int_fun_int_int(plus_plus_int,l),one_one_int)) ) ) ) ).

% of_int_code_if
tff(fact_7047_vebt__predi_Ofixp__induct,axiom,
    ! [P: fun_fu4750890421531948930_nat_o] :
      ( comple7368800650122905596on_nat(partia2080987842261039902Ti_nat(heap_T7048022066654196708on_nat),partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),aTP_Lamm_tp(P))
     => ( aa_fun7244059194882734075_nat_o(P,aTP_Lamm_tq)
       => ( ! [F3: fun_VE8516060933855021849on_nat] :
              ( aa_fun7244059194882734075_nat_o(P,F3)
             => aa_fun7244059194882734075_nat_o(P,aTP_Lamm_tr(F3)) )
         => aa_fun7244059194882734075_nat_o(P,vEBT_vebt_predi) ) ) ) ).

% vebt_predi.fixp_induct
tff(fact_7048_vebt__succi_Ofixp__induct,axiom,
    ! [P: fun_fu4750890421531948930_nat_o] :
      ( comple7368800650122905596on_nat(partia2080987842261039902Ti_nat(heap_T7048022066654196708on_nat),partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),aTP_Lamm_tp(P))
     => ( aa_fun7244059194882734075_nat_o(P,aTP_Lamm_tq)
       => ( ! [F3: fun_VE8516060933855021849on_nat] :
              ( aa_fun7244059194882734075_nat_o(P,F3)
             => aa_fun7244059194882734075_nat_o(P,aTP_Lamm_ts(F3)) )
         => aa_fun7244059194882734075_nat_o(P,vEBT_vebt_succi) ) ) ) ).

% vebt_succi.fixp_induct
tff(fact_7049_compl__le__compl__iff,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Xb)),uminus5710092332889474511et_nat(Y))
    <=> aa_set_nat_o(ord_less_eq_set_nat(Y),Xb) ) ).

% compl_le_compl_iff
tff(fact_7050_ln__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),numeral_numeral_real(aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,Xb) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_tt,Xb)) ) ) ) ).

% ln_series
tff(fact_7051_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_7052_negative__zle,axiom,
    ! [N: nat,M: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zle
tff(fact_7053_negative__zless,axiom,
    ! [N: nat,M: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zless
tff(fact_7054_word__of__int__neg__numeral,axiom,
    ! [Bin: num] : ring_17408606157368542149l_num1(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Bin))) = uminus8244633308260627903l_num1(numera7442385471795722001l_num1(Bin)) ).

% word_of_int_neg_numeral
tff(fact_7055_word__of__int__neg__1,axiom,
    ring_17408606157368542149l_num1(aa_int_int(uminus_uminus_int,one_one_int)) = uminus8244633308260627903l_num1(one_on7727431528512463931l_num1) ).

% word_of_int_neg_1
tff(fact_7056_int__div__minus__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = aa_int_int(uminus_uminus_int,A2) )
      <=> ( B2 = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_7057_powser__zero,axiom,
    ! [Fa: fun_nat_complex] : suminf_complex(aTP_Lamm_tu(Fa)) = aa_nat_complex(Fa,zero_zero_nat) ).

% powser_zero
tff(fact_7058_powser__zero,axiom,
    ! [Fa: fun_nat_real] : suminf_real(aTP_Lamm_tv(Fa)) = aa_nat_real(Fa,zero_zero_nat) ).

% powser_zero
tff(fact_7059_ceiling__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(numeral_numeral_real(A2)),numeral_numeral_real(B2))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A2))),aa_num_int(numeral_numeral_int,B2))) ).

% ceiling_divide_eq_div_numeral
tff(fact_7060_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(numeral_numeral_real(A2)),numeral_numeral_real(B2)))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A2)),aa_num_int(numeral_numeral_int,B2))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_7061_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_7062_uminus__integer__code_I1_J,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% uminus_integer_code(1)
tff(fact_7063_word__neg__numeral__alt,axiom,
    ! [B2: num] : uminus8244633308260627903l_num1(numera7442385471795722001l_num1(B2)) = ring_17408606157368542149l_num1(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,B2))) ).

% word_neg_numeral_alt
tff(fact_7064_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
     => ( ( M = one_one_int )
        | ( M = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_7065_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),N) = one_one_int )
    <=> ( ( ( M = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( N = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_7066_minus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),L) = aa_int_int(uminus_uminus_int,L) ).

% minus_int_code(2)
tff(fact_7067_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
tff(fact_7068_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
tff(fact_7069_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))) ).

% not_int_zless_negative
tff(fact_7070_max__word__not__0,axiom,
    uminus8244633308260627903l_num1(one_on7727431528512463931l_num1) != zero_z3563351764282998399l_num1 ).

% max_word_not_0
tff(fact_7071_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(zero_z3403309356797280102nteger),L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ).

% minus_integer_code(2)
tff(fact_7072_int__cases4,axiom,
    ! [M: int] :
      ( ! [N3: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N3)
     => ~ ! [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3)
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) ) ) ) ).

% int_cases4
tff(fact_7073_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))
    <=> ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_7074_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ~ ! [N3: nat] : K != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) ) ).

% nonpos_int_cases
tff(fact_7075_negative__zle__0,axiom,
    ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))),zero_zero_int) ).

% negative_zle_0
tff(fact_7076_max__word__wrap,axiom,
    ! [Xb: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1(Xb,one_on7727431528512463931l_num1) = zero_z3563351764282998399l_num1 )
     => ( Xb = uminus8244633308260627903l_num1(one_on7727431528512463931l_num1) ) ) ).

% max_word_wrap
tff(fact_7077_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A2)),B2) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2))) ).

% zmod_zminus1_eq_if
tff(fact_7078_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_int_int(uminus_uminus_int,B2)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2)),B2)) ).

% zmod_zminus2_eq_if
tff(fact_7079_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N3: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N3) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3) )
       => ~ ! [N3: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3) ) ) ) ).

% int_cases3
tff(fact_7080_not__zle__0__negative,axiom,
    ! [N: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))) ).

% not_zle_0_negative
tff(fact_7081_negative__zless__0,axiom,
    ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N)))),zero_zero_int) ).

% negative_zless_0
tff(fact_7082_negD,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
     => ? [N3: nat] : Xb = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N3))) ) ).

% negD
tff(fact_7083_verit__less__mono__div__int2,axiom,
    ! [A3: int,B5: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,N))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B5),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),N)) ) ) ).

% verit_less_mono_div_int2
tff(fact_7084_div__eq__minus1,axiom,
    ! [B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),B2) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_7085_ceiling__divide__eq__div,axiom,
    ! [A2: int,B2: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,A2)),aa_int_real(ring_1_of_int_real,B2))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A2)),B2)) ).

% ceiling_divide_eq_div
tff(fact_7086_ceiling__divide__eq__div,axiom,
    ! [A2: int,B2: int] : archim2889992004027027881ng_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,A2),aa_int_rat(ring_1_of_int_rat,B2))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A2)),B2)) ).

% ceiling_divide_eq_div
tff(fact_7087_neg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ~ ! [N3: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N3)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3) ) ) ).

% neg_int_cases
tff(fact_7088_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_7089_zmod__minus1,axiom,
    ! [B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),one_one_int) ) ) ).

% zmod_minus1
tff(fact_7090_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(uminus_uminus_int,B2)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))),one_one_int)) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_7091_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A2)),B2) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))),one_one_int)) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_7092_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q2: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A2,B2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R))
     => ( ( B2 != zero_zero_int )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(uminus_uminus_int,A2),B2),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,
                $ite(R = zero_zero_int,aa_int_int(uminus_uminus_int,Q2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q2)),one_one_int))),
              $ite(R = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),R)))) ) ) ).

% zminus1_lemma
tff(fact_7093_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_7094_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_7095_compl__mono,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Y)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),uminus5710092332889474511et_nat(Xb)) ) ).

% compl_mono
tff(fact_7096_compl__le__swap1,axiom,
    ! [Y: set_nat,Xb: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Y),uminus5710092332889474511et_nat(Xb))
     => aa_set_nat_o(ord_less_eq_set_nat(Xb),uminus5710092332889474511et_nat(Y)) ) ).

% compl_le_swap1
tff(fact_7097_compl__le__swap2,axiom,
    ! [Y: set_nat,Xb: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y)),Xb)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Xb)),Y) ) ).

% compl_le_swap2
tff(fact_7098_int__bit__induct,axiom,
    ! [P: fun_int_o,K: int] :
      ( aa_int_o(P,zero_zero_int)
     => ( aa_int_o(P,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K2: int] :
              ( aa_int_o(P,K2)
             => ( ( K2 != zero_zero_int )
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
         => ( ! [K2: int] :
                ( aa_int_o(P,K2)
               => ( ( K2 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
           => aa_int_o(P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_7099_m1mod2k,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ).

% m1mod2k
tff(fact_7100_sb__dec__lem_H,axiom,
    ! [K: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A2)) ) ).

% sb_dec_lem'
tff(fact_7101_m1mod22k,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),one_one_int) ).

% m1mod22k
tff(fact_7102_sb__inc__lem_H,axiom,
    ! [A2: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))) ) ).

% sb_inc_lem'
tff(fact_7103_sb__dec__lem,axiom,
    ! [K: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A2))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A2)) ) ).

% sb_dec_lem
tff(fact_7104_diff__shunt__var,axiom,
    ! [Xb: assn,Y: assn] :
      ( ( minus_minus_assn(Xb,Y) = bot_bot_assn )
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Xb),Y) ) ).

% diff_shunt_var
tff(fact_7105_diff__shunt__var,axiom,
    ! [Xb: set_real,Y: set_real] :
      ( ( minus_minus_set_real(Xb,Y) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(Xb),Y) ) ).

% diff_shunt_var
tff(fact_7106_diff__shunt__var,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ( minus_minus_set_int(Xb,Y) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(Xb),Y) ) ).

% diff_shunt_var
tff(fact_7107_diff__shunt__var,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ( minus_minus_set_nat(Xb,Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(Xb),Y) ) ).

% diff_shunt_var
tff(fact_7108_suminf__geometric,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => ( suminf_real(aa_real_fun_nat_real(power_power_real,C)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(minus_minus_real(one_one_real),C)) ) ) ).

% suminf_geometric
tff(fact_7109_suminf__geometric,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => ( suminf_complex(aa_com8181001469867462503omplex(power_power_complex,C)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,one_one_complex),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),C)) ) ) ).

% suminf_geometric
tff(fact_7110_minus__one__mod__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,N)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,N))) ).

% minus_one_mod_numeral
tff(fact_7111_one__mod__minus__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,N)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,N)))) ).

% one_mod_minus_numeral
tff(fact_7112_VEBT__internal_Ovebt__memberi_H_Ofixp__induct,axiom,
    ! [P: fun_fu2707964056798801139ap_o_o] :
      ( comple5871978649864577227Heap_o(partia6726927458685305659Ti_nat(heap_Time_Heap_lub_o),partia3290229181235258227Ti_nat(heap_Time_Heap_ord_o),aTP_Lamm_tw(P))
     => ( aa_fun339294776471787884ap_o_o(P,aTP_Lamm_tx)
       => ( ! [F3: fun_VE8181444127144381288Heap_o] :
              ( aa_fun339294776471787884ap_o_o(P,F3)
             => aa_fun339294776471787884ap_o_o(P,aTP_Lamm_ty(F3)) )
         => aa_fun339294776471787884ap_o_o(P,vEBT_V854960066525838166emberi) ) ) ) ).

% VEBT_internal.vebt_memberi'.fixp_induct
tff(fact_7113_one__div__minus__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(one,N))) ).

% one_div_minus_numeral
tff(fact_7114_suminf__zero,axiom,
    suminf_real(aTP_Lamm_tz) = zero_zero_real ).

% suminf_zero
tff(fact_7115_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_iv) = zero_zero_nat ).

% suminf_zero
tff(fact_7116_suminf__zero,axiom,
    suminf_int(aTP_Lamm_ua) = zero_zero_int ).

% suminf_zero
tff(fact_7117_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(M,N))) ).

% minus_numeral_div_numeral
tff(fact_7118_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(M,N))) ).

% numeral_div_minus_numeral
tff(fact_7119_numeral__mod__minus__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,N)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(M,N)))) ).

% numeral_mod_minus_numeral
tff(fact_7120_minus__numeral__mod__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,N)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(M,N))) ).

% minus_numeral_mod_numeral
tff(fact_7121_minus__one__div__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(one,N))) ).

% minus_one_div_numeral
tff(fact_7122_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_7123_bot__option__def,axiom,
    bot_bot_option_num = none_num ).

% bot_option_def
tff(fact_7124_bot__option__def,axiom,
    bot_bot_option_nat = none_nat ).

% bot_option_def
tff(fact_7125_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_7126_bot__empty__eq2,axiom,
    ! [X3: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(bot_bo3657453288633029378_num_o,X3),Xa)
    <=> member7279096912039735102um_num(aa_num3156681643174609981um_num(product_Pair_num_num(X3),Xa),bot_bo9056780473022590049um_num) ) ).

% bot_empty_eq2
tff(fact_7127_bot__empty__eq2,axiom,
    ! [X3: nat,Xa: produc4813437837504472865T_VEBT] :
      ( aa_Pro3461009536670225267VEBT_o(aa_nat6863640065291604488VEBT_o(bot_bo9198892711125070427VEBT_o,X3),Xa)
    <=> member306291179834725981T_VEBT(aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(X3),Xa),bot_bo9115540109607619856T_VEBT) ) ).

% bot_empty_eq2
tff(fact_7128_bot__empty__eq2,axiom,
    ! [X3: nat,Xa: num] :
      ( aa_num_o(aa_nat_fun_num_o(bot_bo1877178017081890764_num_o,X3),Xa)
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(X3),Xa),bot_bo7038385379056416535at_num) ) ).

% bot_empty_eq2
tff(fact_7129_bot__empty__eq2,axiom,
    ! [X3: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X3),Xa)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X3),Xa),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_7130_bot__empty__eq2,axiom,
    ! [X3: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(bot_bo428638681919004382_int_o,X3),Xa)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_7131_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R: int] :
      aa_int_int(aa_int_fun_int_int(adjust_mod,L),R) = $ite(R = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),R)) ).

% Divides.adjust_mod_def
tff(fact_7132_vebt__memberi_Ofixp__induct,axiom,
    ! [P: fun_fu8886734894937297424ap_o_o] :
      ( comple8003709893544916456Heap_o(partia5551857090987368152Ti_nat(heap_Time_Heap_lub_o),partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),aTP_Lamm_ub(P))
     => ( aa_fun4656092467790859897ap_o_o(P,aTP_Lamm_uc)
       => ( ! [F3: fun_VE7134199479003659425Heap_o] :
              ( aa_fun4656092467790859897ap_o_o(P,F3)
             => aa_fun4656092467790859897ap_o_o(P,aTP_Lamm_ud(F3)) )
         => aa_fun4656092467790859897ap_o_o(P,vEBT_vebt_memberi) ) ) ) ).

% vebt_memberi.fixp_induct
tff(fact_7133_pi__series,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_ue) ).

% pi_series
tff(fact_7134_integer__of__int__code,axiom,
    ! [K: int] :
      aa_int_Code_integer(code_integer_of_int,K) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,K))),
        $ite(
          K = zero_zero_int,
          zero_z3403309356797280102nteger,
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(l),one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_7135_dbl__dec__simps_I4_J,axiom,
    neg_nu93272222329896523l_num1(uminus8244633308260627903l_num1(one_on7727431528512463931l_num1)) = uminus8244633308260627903l_num1(numera7442385471795722001l_num1(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7136_dbl__dec__simps_I4_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7137_dbl__dec__simps_I4_J,axiom,
    neg_nu965353292909893953uint32(uminus_uminus_uint32(one_one_uint32)) = uminus_uminus_uint32(numera9087168376688890119uint32(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7138_dbl__dec__simps_I4_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7139_dbl__dec__simps_I4_J,axiom,
    neg_nu3179335615603231917ec_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7140_dbl__dec__simps_I4_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_7141_dbl__dec__simps_I3_J,axiom,
    neg_nu965353292909893953uint32(one_one_uint32) = one_one_uint32 ).

% dbl_dec_simps(3)
tff(fact_7142_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_7143_dbl__dec__simps_I3_J,axiom,
    neg_nu3179335615603231917ec_rat(one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_7144_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_7145_dbl__dec__simps_I2_J,axiom,
    neg_nu93272222329896523l_num1(zero_z3563351764282998399l_num1) = uminus8244633308260627903l_num1(one_on7727431528512463931l_num1) ).

% dbl_dec_simps(2)
tff(fact_7146_dbl__dec__simps_I2_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,zero_zero_complex) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_dec_simps(2)
tff(fact_7147_dbl__dec__simps_I2_J,axiom,
    neg_nu965353292909893953uint32(zero_zero_uint32) = uminus_uminus_uint32(one_one_uint32) ).

% dbl_dec_simps(2)
tff(fact_7148_dbl__dec__simps_I2_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,zero_zero_real) = aa_real_real(uminus_uminus_real,one_one_real) ).

% dbl_dec_simps(2)
tff(fact_7149_dbl__dec__simps_I2_J,axiom,
    neg_nu3179335615603231917ec_rat(zero_zero_rat) = uminus_uminus_rat(one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_7150_dbl__dec__simps_I2_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_dec_simps(2)
tff(fact_7151_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
tff(fact_7152_zero__integer__def,axiom,
    zero_z3403309356797280102nteger = aa_int_Code_integer(code_integer_of_int,zero_zero_int) ).

% zero_integer_def
tff(fact_7153_less__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xaa)),aa_int_Code_integer(code_integer_of_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xaa),Xb) ) ).

% less_integer.abs_eq
tff(fact_7154_one__integer__def,axiom,
    one_one_Code_integer = aa_int_Code_integer(code_integer_of_int,one_one_int) ).

% one_integer_def
tff(fact_7155_pi__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),pi) ).

% pi_gt_zero
tff(fact_7156_pi__not__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,pi),zero_zero_real) ).

% pi_not_less_zero
tff(fact_7157_pi__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),pi) ).

% pi_ge_zero
tff(fact_7158_less__eq__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(code_integer_of_int,Xaa)),aa_int_Code_integer(code_integer_of_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xaa),Xb) ) ).

% less_eq_integer.abs_eq
tff(fact_7159_divide__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(aa_int_Code_integer(code_integer_of_int,Xaa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),Xb)) ).

% divide_integer.abs_eq
tff(fact_7160_pi__less__4,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% pi_less_4
tff(fact_7161_pi__ge__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),pi) ).

% pi_ge_two
tff(fact_7162_pi__half__neq__two,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))) != numeral_numeral_real(aa_num_num(bit0,one)) ).

% pi_half_neq_two
tff(fact_7163_pi__half__neq__zero,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_7164_pi__half__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% pi_half_less_two
tff(fact_7165_pi__half__le__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% pi_half_le_two
tff(fact_7166_dbl__dec__def,axiom,
    ! [Xb: uint32] : neg_nu965353292909893953uint32(Xb) = minus_minus_uint32(plus_plus_uint32(Xb,Xb),one_one_uint32) ).

% dbl_dec_def
tff(fact_7167_dbl__dec__def,axiom,
    ! [Xb: real] : aa_real_real(neg_nu6075765906172075777c_real,Xb) = aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb)),one_one_real) ).

% dbl_dec_def
tff(fact_7168_dbl__dec__def,axiom,
    ! [Xb: rat] : neg_nu3179335615603231917ec_rat(Xb) = minus_minus_rat(plus_plus_rat(Xb,Xb),one_one_rat) ).

% dbl_dec_def
tff(fact_7169_dbl__dec__def,axiom,
    ! [Xb: int] : aa_int_int(neg_nu3811975205180677377ec_int,Xb) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb)),one_one_int) ).

% dbl_dec_def
tff(fact_7170_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% pi_half_gt_zero
tff(fact_7171_pi__half__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% pi_half_ge_zero
tff(fact_7172_m2pi__less__pi,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))),pi) ).

% m2pi_less_pi
tff(fact_7173_minus__pi__half__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),zero_zero_real) ).

% minus_pi_half_less_zero
tff(fact_7174_heap_Omono__body__fixp,axiom,
    ! [F4: fun_fu411293875173739919on_nat] :
      ( ! [X2: produc3881548065746020326Ti_nat] : comple7427593846244870586on_nat(partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),heap_T7875578835903804844on_nat,aa_Pro8654539709784947602on_nat(aTP_Lamm_uf(F4),X2))
     => ( comple6035428378238101126on_nat(partia2080987842261039902Ti_nat(heap_T7048022066654196708on_nat),partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),F4) = aa_fun943833532922116944on_nat(F4,comple6035428378238101126on_nat(partia2080987842261039902Ti_nat(heap_T7048022066654196708on_nat),partia8535946980722491222Ti_nat(heap_T7875578835903804844on_nat),F4)) ) ) ).

% heap.mono_body_fixp
tff(fact_7175_heap_Omono__body__fixp,axiom,
    ! [F4: fun_fu2426956696497423235Heap_o] :
      ( ! [X2: produc3881548065746020326Ti_nat] : comple6933591415923782038Heap_o(partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aa_Pro5165804456746783526Heap_o(aTP_Lamm_ug(F4),X2))
     => ( comple8693199471913871838Heap_o(partia5551857090987368152Ti_nat(heap_Time_Heap_lub_o),partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),F4) = aa_fun5049358401508434812Heap_o(F4,comple8693199471913871838Heap_o(partia5551857090987368152Ti_nat(heap_Time_Heap_lub_o),partia2925774515620677392Ti_nat(heap_Time_Heap_ord_o),F4)) ) ) ).

% heap.mono_body_fixp
tff(fact_7176_heap_Omono__body__fixp,axiom,
    ! [F4: fun_fu1151686773700218227_VEBTi] :
      ( ! [X2: produc3960855945107176009Ti_nat] : comple422786390015765981_VEBTi(partia1868168049876374393Ti_nat(heap_T7173139186834293313_VEBTi),heap_T7173139186834293313_VEBTi,aa_Pro7504462699960347282_VEBTi(aTP_Lamm_uh(F4),X2))
     => ( comple5861331542469577976_VEBTi(partia7782936097874681665Ti_nat(heap_T3112222404744780921_VEBTi),partia1868168049876374393Ti_nat(heap_T7173139186834293313_VEBTi),F4) = aa_fun5250146550359294516_VEBTi(F4,comple5861331542469577976_VEBTi(partia7782936097874681665Ti_nat(heap_T3112222404744780921_VEBTi),partia1868168049876374393Ti_nat(heap_T7173139186834293313_VEBTi),F4)) ) ) ).

% heap.mono_body_fixp
tff(fact_7177_heap_Omono__body__fixp,axiom,
    ! [F4: fun_fu3620910142932681on_nat] :
      ( ! [X2: produc3960855945107176009Ti_nat] : comple4513117127444775325on_nat(partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),heap_T7875578835903804844on_nat,aa_Pro1535628157467025618on_nat(aTP_Lamm_ui(F4),X2))
     => ( comple4104493882897929443on_nat(partia6039416512482706817Ti_nat(heap_T7048022066654196708on_nat),partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),F4) = aa_fun7861206497237518090on_nat(F4,comple4104493882897929443on_nat(partia6039416512482706817Ti_nat(heap_T7048022066654196708on_nat),partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),F4)) ) ) ).

% heap.mono_body_fixp
tff(fact_7178_heap_Omono__body__fixp,axiom,
    ! [F4: fun_fu8308473583445096899Heap_o] :
      ( ! [X2: produc3960855945107176009Ti_nat] : comple5241396456474302457Heap_o(partia3290229181235258227Ti_nat(heap_Time_Heap_ord_o),heap_Time_Heap_ord_o,aa_Pro9147008720614807654Heap_o(aTP_Lamm_uj(F4),X2))
     => ( comple5362527178602408641Heap_o(partia6726927458685305659Ti_nat(heap_Time_Heap_lub_o),partia3290229181235258227Ti_nat(heap_Time_Heap_ord_o),F4) = aa_fun55487300827498300Heap_o(F4,comple5362527178602408641Heap_o(partia6726927458685305659Ti_nat(heap_Time_Heap_lub_o),partia3290229181235258227Ti_nat(heap_Time_Heap_ord_o),F4)) ) ) ).

% heap.mono_body_fixp
tff(fact_7179_sin__cos__npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),pi)),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),N) ).

% sin_cos_npi
tff(fact_7180_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_7181_signed__take__bit__rec,axiom,
    ! [N: nat,A2: word_N3645301735248828278l_num1] :
      bit_ri1375673916561920181l_num1(N,A2) = $ite(N = zero_zero_nat,uminus8244633308260627903l_num1(modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one)))),plus_p361126936061061375l_num1(modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),bit_ri1375673916561920181l_num1(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide1791077408188789448l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_7182_signed__take__bit__rec,axiom,
    ! [N: nat,A2: code_integer] :
      aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N),A2) = $ite(N = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_7183_signed__take__bit__rec,axiom,
    ! [N: nat,A2: uint32] :
      bit_ri6224792872505173163uint32(N,A2) = $ite(N = zero_zero_nat,uminus_uminus_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one)))),plus_plus_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),bit_ri6224792872505173163uint32(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_7184_signed__take__bit__rec,axiom,
    ! [N: nat,A2: int] :
      aa_int_int(bit_ri631733984087533419it_int(N),A2) = $ite(N = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_7185_dbl__simps_I4_J,axiom,
    neg_nu7865238048354675525l_num1(uminus8244633308260627903l_num1(one_on7727431528512463931l_num1)) = uminus8244633308260627903l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_7186_dbl__simps_I4_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_7187_dbl__simps_I4_J,axiom,
    neg_nu5314729912787363643uint32(uminus_uminus_uint32(one_one_uint32)) = uminus_uminus_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_7188_dbl__simps_I4_J,axiom,
    aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_7189_dbl__simps_I4_J,axiom,
    neg_numeral_dbl_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_7190_dbl__simps_I4_J,axiom,
    aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_7191_cos__pi__eq__zero,axiom,
    ! [M: nat] : aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))))),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_7192_abs__abs,axiom,
    ! [A2: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A2)) = aa_real_real(abs_abs_real,A2) ).

% abs_abs
tff(fact_7193_abs__abs,axiom,
    ! [A2: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A2)) = aa_int_int(abs_abs_int,A2) ).

% abs_abs
tff(fact_7194_abs__abs,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2) ).

% abs_abs
tff(fact_7195_abs__abs,axiom,
    ! [A2: rat] : abs_abs_rat(abs_abs_rat(A2)) = abs_abs_rat(A2) ).

% abs_abs
tff(fact_7196_abs__0,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_7197_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_7198_abs__0,axiom,
    abs_abs_rat(zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_7199_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_7200_abs__0__eq,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2) )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_7201_abs__0__eq,axiom,
    ! [A2: real] :
      ( ( zero_zero_real = aa_real_real(abs_abs_real,A2) )
    <=> ( A2 = zero_zero_real ) ) ).

% abs_0_eq
tff(fact_7202_abs__0__eq,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat = abs_abs_rat(A2) )
    <=> ( A2 = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_7203_abs__0__eq,axiom,
    ! [A2: int] :
      ( ( zero_zero_int = aa_int_int(abs_abs_int,A2) )
    <=> ( A2 = zero_zero_int ) ) ).

% abs_0_eq
tff(fact_7204_abs__eq__0,axiom,
    ! [A2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2) = zero_z3403309356797280102nteger )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_7205_abs__eq__0,axiom,
    ! [A2: real] :
      ( ( aa_real_real(abs_abs_real,A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% abs_eq_0
tff(fact_7206_abs__eq__0,axiom,
    ! [A2: rat] :
      ( ( abs_abs_rat(A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_7207_abs__eq__0,axiom,
    ! [A2: int] :
      ( ( aa_int_int(abs_abs_int,A2) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% abs_eq_0
tff(fact_7208_abs__zero,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_7209_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_7210_abs__zero,axiom,
    abs_abs_rat(zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_7211_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_7212_abs__numeral,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,numera6620942414471956472nteger(N)) = numera6620942414471956472nteger(N) ).

% abs_numeral
tff(fact_7213_abs__numeral,axiom,
    ! [N: num] : aa_real_real(abs_abs_real,numeral_numeral_real(N)) = numeral_numeral_real(N) ).

% abs_numeral
tff(fact_7214_abs__numeral,axiom,
    ! [N: num] : abs_abs_rat(numeral_numeral_rat(N)) = numeral_numeral_rat(N) ).

% abs_numeral
tff(fact_7215_abs__numeral,axiom,
    ! [N: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,N)) = aa_num_int(numeral_numeral_int,N) ).

% abs_numeral
tff(fact_7216_abs__1,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_7217_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_7218_abs__1,axiom,
    abs_abs_rat(one_one_rat) = one_one_rat ).

% abs_1
tff(fact_7219_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_7220_abs__mult__self__eq,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),A2) ).

% abs_mult_self_eq
tff(fact_7221_abs__mult__self__eq,axiom,
    ! [A2: real] : aa_real_real(times_times_real(aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,A2)) = aa_real_real(times_times_real(A2),A2) ).

% abs_mult_self_eq
tff(fact_7222_abs__mult__self__eq,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(abs_abs_rat(A2)),abs_abs_rat(A2)) = aa_rat_rat(times_times_rat(A2),A2) ).

% abs_mult_self_eq
tff(fact_7223_abs__mult__self__eq,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,A2)) = aa_int_int(aa_int_fun_int_int(times_times_int,A2),A2) ).

% abs_mult_self_eq
tff(fact_7224_abs__divide,axiom,
    ! [A2: real,B2: real] : aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(A2),B2)) = aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2)) ).

% abs_divide
tff(fact_7225_abs__divide,axiom,
    ! [A2: rat,B2: rat] : abs_abs_rat(divide_divide_rat(A2,B2)) = divide_divide_rat(abs_abs_rat(A2),abs_abs_rat(B2)) ).

% abs_divide
tff(fact_7226_abs__minus,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2) ).

% abs_minus
tff(fact_7227_abs__minus,axiom,
    ! [A2: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)) = aa_complex_complex(abs_abs_complex,A2) ).

% abs_minus
tff(fact_7228_abs__minus,axiom,
    ! [A2: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A2)) = aa_real_real(abs_abs_real,A2) ).

% abs_minus
tff(fact_7229_abs__minus,axiom,
    ! [A2: rat] : abs_abs_rat(uminus_uminus_rat(A2)) = abs_abs_rat(A2) ).

% abs_minus
tff(fact_7230_abs__minus,axiom,
    ! [A2: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A2)) = aa_int_int(abs_abs_int,A2) ).

% abs_minus
tff(fact_7231_signed__take__bit__of__0,axiom,
    ! [N: nat] : bit_ri1375673916561920181l_num1(N,zero_z3563351764282998399l_num1) = zero_z3563351764282998399l_num1 ).

% signed_take_bit_of_0
tff(fact_7232_signed__take__bit__of__0,axiom,
    ! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(N),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_7233_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(abs_abs_real,M)),K)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,M),K) ) ).

% abs_dvd_iff
tff(fact_7234_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(abs_abs_int,M)),K)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),K) ) ).

% abs_dvd_iff
tff(fact_7235_abs__dvd__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,M)),K)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),K) ) ).

% abs_dvd_iff
tff(fact_7236_abs__dvd__iff,axiom,
    ! [M: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,abs_abs_rat(M)),K)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),K) ) ).

% abs_dvd_iff
tff(fact_7237_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,M),aa_real_real(abs_abs_real,K))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,M),K) ) ).

% dvd_abs_iff
tff(fact_7238_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),aa_int_int(abs_abs_int,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),K) ) ).

% dvd_abs_iff
tff(fact_7239_dvd__abs__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),K) ) ).

% dvd_abs_iff
tff(fact_7240_dvd__abs__iff,axiom,
    ! [M: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),abs_abs_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),K) ) ).

% dvd_abs_iff
tff(fact_7241_abs__of__nat,axiom,
    ! [N: nat] : abs_abs_rat(aa_nat_rat(semiri681578069525770553at_rat,N)) = aa_nat_rat(semiri681578069525770553at_rat,N) ).

% abs_of_nat
tff(fact_7242_abs__of__nat,axiom,
    ! [N: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(semiri5074537144036343181t_real,N) ).

% abs_of_nat
tff(fact_7243_abs__of__nat,axiom,
    ! [N: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,N)) = aa_nat_int(semiri1314217659103216013at_int,N) ).

% abs_of_nat
tff(fact_7244_abs__of__nat,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,N)) = aa_nat_Code_integer(semiri4939895301339042750nteger,N) ).

% abs_of_nat
tff(fact_7245_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_7246_dbl__simps_I2_J,axiom,
    neg_nu7865238048354675525l_num1(zero_z3563351764282998399l_num1) = zero_z3563351764282998399l_num1 ).

% dbl_simps(2)
tff(fact_7247_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_7248_dbl__simps_I2_J,axiom,
    neg_numeral_dbl_rat(zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_7249_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_7250_abs__of__nonneg,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2) = A2 ) ) ).

% abs_of_nonneg
tff(fact_7251_abs__of__nonneg,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_real(abs_abs_real,A2) = A2 ) ) ).

% abs_of_nonneg
tff(fact_7252_abs__of__nonneg,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( abs_abs_rat(A2) = A2 ) ) ).

% abs_of_nonneg
tff(fact_7253_abs__of__nonneg,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_int(abs_abs_int,A2) = A2 ) ) ).

% abs_of_nonneg
tff(fact_7254_abs__le__self__iff,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),A2)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2) ) ).

% abs_le_self_iff
tff(fact_7255_abs__le__self__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ).

% abs_le_self_iff
tff(fact_7256_abs__le__self__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A2)),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ).

% abs_le_self_iff
tff(fact_7257_abs__le__self__iff,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),A2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ).

% abs_le_self_iff
tff(fact_7258_abs__le__zero__iff,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),zero_z3403309356797280102nteger)
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_7259_abs__le__zero__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),zero_zero_real)
    <=> ( A2 = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_7260_abs__le__zero__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A2)),zero_zero_rat)
    <=> ( A2 = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_7261_abs__le__zero__iff,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),zero_zero_int)
    <=> ( A2 = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_7262_zero__less__abs__iff,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2))
    <=> ( A2 != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_7263_zero__less__abs__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A2))
    <=> ( A2 != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_7264_zero__less__abs__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),abs_abs_rat(A2))
    <=> ( A2 != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_7265_zero__less__abs__iff,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A2))
    <=> ( A2 != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_7266_abs__neg__numeral,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(N) ).

% abs_neg_numeral
tff(fact_7267_abs__neg__numeral,axiom,
    ! [N: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = numeral_numeral_real(N) ).

% abs_neg_numeral
tff(fact_7268_abs__neg__numeral,axiom,
    ! [N: num] : abs_abs_rat(uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(N) ).

% abs_neg_numeral
tff(fact_7269_abs__neg__numeral,axiom,
    ! [N: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,N) ).

% abs_neg_numeral
tff(fact_7270_abs__neg__one,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_7271_abs__neg__one,axiom,
    aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,one_one_real)) = one_one_real ).

% abs_neg_one
tff(fact_7272_abs__neg__one,axiom,
    abs_abs_rat(uminus_uminus_rat(one_one_rat)) = one_one_rat ).

% abs_neg_one
tff(fact_7273_abs__neg__one,axiom,
    aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,one_one_int)) = one_one_int ).

% abs_neg_one
tff(fact_7274_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_7275_abs__power__minus,axiom,
    ! [A2: code_integer,N: nat] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),N)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) ).

% abs_power_minus
tff(fact_7276_abs__power__minus,axiom,
    ! [A2: real,N: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),N)) = aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) ).

% abs_power_minus
tff(fact_7277_abs__power__minus,axiom,
    ! [A2: rat,N: nat] : abs_abs_rat(aa_nat_rat(power_power_rat(uminus_uminus_rat(A2)),N)) = abs_abs_rat(aa_nat_rat(power_power_rat(A2),N)) ).

% abs_power_minus
tff(fact_7278_abs__power__minus,axiom,
    ! [A2: int,N: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),N)) = aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A2),N)) ).

% abs_power_minus
tff(fact_7279_signed__take__bit__Suc__1,axiom,
    ! [N: nat] : bit_ri6224792872505173163uint32(aa_nat_nat(suc,N),one_one_uint32) = one_one_uint32 ).

% signed_take_bit_Suc_1
tff(fact_7280_signed__take__bit__Suc__1,axiom,
    ! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_7281_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : bit_ri6224792872505173163uint32(aa_num_nat(numeral_numeral_nat,K),one_one_uint32) = one_one_uint32 ).

% signed_take_bit_numeral_of_1
tff(fact_7282_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,K)),one_one_int) = one_one_int ).

% signed_take_bit_numeral_of_1
tff(fact_7283_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] : bit_ri6224792872505173163uint32(N,uminus_uminus_uint32(one_one_uint32)) = uminus_uminus_uint32(one_one_uint32) ).

% signed_take_bit_of_minus_1
tff(fact_7284_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% signed_take_bit_of_minus_1
tff(fact_7285_sin__pi,axiom,
    aa_real_real(sin_real,pi) = zero_zero_real ).

% sin_pi
tff(fact_7286_dbl__simps_I5_J,axiom,
    ! [K: num] : neg_nu7865238048354675525l_num1(numera7442385471795722001l_num1(K)) = numera7442385471795722001l_num1(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_7287_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,numeral_numeral_real(K)) = numeral_numeral_real(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_7288_dbl__simps_I5_J,axiom,
    ! [K: num] : neg_numeral_dbl_rat(numeral_numeral_rat(K)) = numeral_numeral_rat(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_7289_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_7290_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu7865238048354675525l_num1(uminus8244633308260627903l_num1(numera7442385471795722001l_num1(K))) = uminus8244633308260627903l_num1(neg_nu7865238048354675525l_num1(numera7442385471795722001l_num1(K))) ).

% dbl_simps(1)
tff(fact_7291_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K))) ).

% dbl_simps(1)
tff(fact_7292_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu5314729912787363643uint32(uminus_uminus_uint32(numera9087168376688890119uint32(K))) = uminus_uminus_uint32(neg_nu5314729912787363643uint32(numera9087168376688890119uint32(K))) ).

% dbl_simps(1)
tff(fact_7293_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,numeral_numeral_real(K))) ).

% dbl_simps(1)
tff(fact_7294_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)
tff(fact_7295_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_simps(1)
tff(fact_7296_divide__le__0__abs__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A2),aa_real_real(abs_abs_real,B2))),zero_zero_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
        | ( B2 = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_7297_divide__le__0__abs__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,divide_divide_rat(A2,abs_abs_rat(B2))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
        | ( B2 = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_7298_zero__le__divide__abs__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(A2),aa_real_real(abs_abs_real,B2)))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
        | ( B2 = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_7299_zero__le__divide__abs__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),divide_divide_rat(A2,abs_abs_rat(B2)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
        | ( B2 = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_7300_abs__of__nonpos,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger)
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) ) ) ).

% abs_of_nonpos
tff(fact_7301_abs__of__nonpos,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A2) = aa_real_real(uminus_uminus_real,A2) ) ) ).

% abs_of_nonpos
tff(fact_7302_abs__of__nonpos,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
     => ( abs_abs_rat(A2) = uminus_uminus_rat(A2) ) ) ).

% abs_of_nonpos
tff(fact_7303_abs__of__nonpos,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A2) = aa_int_int(uminus_uminus_int,A2) ) ) ).

% abs_of_nonpos
tff(fact_7304_artanh__minus__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,Xb)) ) ) ).

% artanh_minus_real
tff(fact_7305_zero__less__power__abs__iff,axiom,
    ! [A2: code_integer,N: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),N))
    <=> ( ( A2 != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7306_zero__less__power__abs__iff,axiom,
    ! [A2: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),N))
    <=> ( ( A2 != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7307_zero__less__power__abs__iff,axiom,
    ! [A2: rat,N: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(abs_abs_rat(A2)),N))
    <=> ( ( A2 != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7308_zero__less__power__abs__iff,axiom,
    ! [A2: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A2)),N))
    <=> ( ( A2 != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_7309_abs__power2,axiom,
    ! [A2: rat] : abs_abs_rat(aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_7310_abs__power2,axiom,
    ! [A2: int] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_7311_abs__power2,axiom,
    ! [A2: real] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_7312_abs__power2,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_7313_power2__abs,axiom,
    ! [A2: rat] : aa_nat_rat(power_power_rat(abs_abs_rat(A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_7314_power2__abs,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_7315_power2__abs,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_7316_power2__abs,axiom,
    ! [A2: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_7317_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_bit0
tff(fact_7318_sin__cos__squared__add3,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Xb))),aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Xb))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_7319_sin__npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_7320_sin__npi2,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,N))) = zero_zero_real ).

% sin_npi2
tff(fact_7321_dbl__simps_I3_J,axiom,
    neg_nu5314729912787363643uint32(one_one_uint32) = numera9087168376688890119uint32(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_7322_dbl__simps_I3_J,axiom,
    neg_nu7865238048354675525l_num1(one_on7727431528512463931l_num1) = numera7442385471795722001l_num1(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_7323_dbl__simps_I3_J,axiom,
    aa_real_real(neg_numeral_dbl_real,one_one_real) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_7324_dbl__simps_I3_J,axiom,
    neg_numeral_dbl_rat(one_one_rat) = numeral_numeral_rat(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_7325_dbl__simps_I3_J,axiom,
    aa_int_int(neg_numeral_dbl_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_7326_sin__npi__int,axiom,
    ! [N: int] : aa_real_real(sin_real,aa_real_real(times_times_real(pi),aa_int_real(ring_1_of_int_real,N))) = zero_zero_real ).

% sin_npi_int
tff(fact_7327_power__even__abs__numeral,axiom,
    ! [W: num,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_rat(power_power_rat(abs_abs_rat(A2)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_7328_power__even__abs__numeral,axiom,
    ! [W: num,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A2)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_7329_power__even__abs__numeral,axiom,
    ! [W: num,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_7330_power__even__abs__numeral,axiom,
    ! [W: num,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_7331_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_7332_signed__take__bit__0,axiom,
    ! [A2: word_N3645301735248828278l_num1] : bit_ri1375673916561920181l_num1(zero_zero_nat,A2) = uminus8244633308260627903l_num1(modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_7333_signed__take__bit__0,axiom,
    ! [A2: code_integer] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(zero_zero_nat),A2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_7334_signed__take__bit__0,axiom,
    ! [A2: uint32] : bit_ri6224792872505173163uint32(zero_zero_nat,A2) = uminus_uminus_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_7335_signed__take__bit__0,axiom,
    ! [A2: int] : aa_int_int(bit_ri631733984087533419it_int(zero_zero_nat),A2) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_7336_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_7337_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_7338_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_7339_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) = one_one_real ).

% sin_pi_half
tff(fact_7340_cos__periodic,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,Xb) ).

% cos_periodic
tff(fact_7341_sin__periodic,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,Xb) ).

% sin_periodic
tff(fact_7342_cos__2pi__minus,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)),Xb)) = aa_real_real(cos_real,Xb) ).

% cos_2pi_minus
tff(fact_7343_sin__cos__squared__add,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_7344_sin__cos__squared__add,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_7345_sin__cos__squared__add2,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_7346_sin__cos__squared__add2,axiom,
    ! [Xb: complex] : aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_7347_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_7348_sin__2npi,axiom,
    ! [N: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_7349_cos__2npi,axiom,
    ! [N: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)) = one_one_real ).

% cos_2npi
tff(fact_7350_sin__2pi__minus,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_2pi_minus
tff(fact_7351_sin__int__2pin,axiom,
    ! [N: int] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,N))) = zero_zero_real ).

% sin_int_2pin
tff(fact_7352_cos__int__2pin,axiom,
    ! [N: int] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,N))) = one_one_real ).

% cos_int_2pin
tff(fact_7353_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(divide_divide_real(numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_7354_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_7355_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(divide_divide_real(numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_7356_cos__npi__int,axiom,
    ! [N: int] :
      aa_real_real(cos_real,aa_real_real(times_times_real(pi),aa_int_real(ring_1_of_int_real,N))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% cos_npi_int
tff(fact_7357_abs__one,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_7358_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_7359_abs__one,axiom,
    abs_abs_rat(one_one_rat) = one_one_rat ).

% abs_one
tff(fact_7360_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_7361_abs__mult,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B2)) ).

% abs_mult
tff(fact_7362_abs__mult,axiom,
    ! [A2: real,B2: real] : aa_real_real(abs_abs_real,aa_real_real(times_times_real(A2),B2)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2)) ).

% abs_mult
tff(fact_7363_abs__mult,axiom,
    ! [A2: rat,B2: rat] : abs_abs_rat(aa_rat_rat(times_times_rat(A2),B2)) = aa_rat_rat(times_times_rat(abs_abs_rat(A2)),abs_abs_rat(B2)) ).

% abs_mult
tff(fact_7364_abs__mult,axiom,
    ! [A2: int,B2: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2)) ).

% abs_mult
tff(fact_7365_power__abs,axiom,
    ! [A2: rat,N: nat] : abs_abs_rat(aa_nat_rat(power_power_rat(A2),N)) = aa_nat_rat(power_power_rat(abs_abs_rat(A2)),N) ).

% power_abs
tff(fact_7366_power__abs,axiom,
    ! [A2: int,N: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A2),N)) = aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A2)),N) ).

% power_abs
tff(fact_7367_power__abs,axiom,
    ! [A2: real,N: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),N) ).

% power_abs
tff(fact_7368_power__abs,axiom,
    ! [A2: code_integer,N: nat] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),N) ).

% power_abs
tff(fact_7369_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( aa_real_real(abs_abs_real,L) = aa_real_real(abs_abs_real,K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),K) ) ).

% dvd_if_abs_eq
tff(fact_7370_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( aa_int_int(abs_abs_int,L) = aa_int_int(abs_abs_int,K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K) ) ).

% dvd_if_abs_eq
tff(fact_7371_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),K) ) ).

% dvd_if_abs_eq
tff(fact_7372_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( abs_abs_rat(L) = abs_abs_rat(K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),K) ) ).

% dvd_if_abs_eq
tff(fact_7373_sin__zero__abs__cos__one,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(cos_real,Xb)) = one_one_real ) ) ).

% sin_zero_abs_cos_one
tff(fact_7374_cos__one__sin__zero,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
     => ( aa_real_real(sin_real,Xb) = zero_zero_real ) ) ).

% cos_one_sin_zero
tff(fact_7375_abs__ge__self,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(abs_abs_real,A2)) ).

% abs_ge_self
tff(fact_7376_abs__ge__self,axiom,
    ! [A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)) ).

% abs_ge_self
tff(fact_7377_abs__ge__self,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),abs_abs_rat(A2)) ).

% abs_ge_self
tff(fact_7378_abs__ge__self,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(abs_abs_int,A2)) ).

% abs_ge_self
tff(fact_7379_abs__le__D1,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% abs_le_D1
tff(fact_7380_abs__le__D1,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2) ) ).

% abs_le_D1
tff(fact_7381_abs__le__D1,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A2)),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% abs_le_D1
tff(fact_7382_abs__le__D1,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% abs_le_D1
tff(fact_7383_abs__eq__0__iff,axiom,
    ! [A2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2) = zero_z3403309356797280102nteger )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_7384_abs__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( aa_real_real(abs_abs_real,A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% abs_eq_0_iff
tff(fact_7385_abs__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( abs_abs_rat(A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_7386_abs__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( aa_int_int(abs_abs_int,A2) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% abs_eq_0_iff
tff(fact_7387_signed__take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_int_int(bit_ri631733984087533419it_int(N),L))) = aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).

% signed_take_bit_mult
tff(fact_7388_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_int_int(bit_ri631733984087533419it_int(N),L))) = aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ).

% signed_take_bit_add
tff(fact_7389_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,aa_int_int(bit_ri631733984087533419it_int(N),K))) = aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(uminus_uminus_int,K)) ).

% signed_take_bit_minus
tff(fact_7390_abs__eq__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% abs_eq_iff
tff(fact_7391_abs__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(abs_abs_real,Xb) = aa_real_real(abs_abs_real,Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% abs_eq_iff
tff(fact_7392_abs__eq__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( abs_abs_rat(Xb) = abs_abs_rat(Y) )
    <=> ( ( Xb = Y )
        | ( Xb = uminus_uminus_rat(Y) ) ) ) ).

% abs_eq_iff
tff(fact_7393_abs__eq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_int_int(abs_abs_int,Xb) = aa_int_int(abs_abs_int,Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% abs_eq_iff
tff(fact_7394_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_int_int(bit_ri631733984087533419it_int(N),L))) = aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% signed_take_bit_diff
tff(fact_7395_sin__zero__norm__cos__one,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(cos_real,Xb)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_7396_sin__zero__norm__cos__one,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(sin_complex,Xb) = zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(cos_complex,Xb)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_7397_sin__zero__pi__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),pi)
     => ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_7398_sin__double,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),Xb)) = aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,Xb))),aa_real_real(cos_real,Xb)) ).

% sin_double
tff(fact_7399_sincos__principal__value,axiom,
    ! [Xb: real] :
    ? [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y2)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),pi)
      & ( aa_real_real(sin_real,Y2) = aa_real_real(sin_real,Xb) )
      & ( aa_real_real(cos_real,Y2) = aa_real_real(cos_real,Xb) ) ) ).

% sincos_principal_value
tff(fact_7400_abs__ge__zero,axiom,
    ! [A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)) ).

% abs_ge_zero
tff(fact_7401_abs__ge__zero,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A2)) ).

% abs_ge_zero
tff(fact_7402_abs__ge__zero,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),abs_abs_rat(A2)) ).

% abs_ge_zero
tff(fact_7403_abs__ge__zero,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A2)) ).

% abs_ge_zero
tff(fact_7404_abs__of__pos,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2) = A2 ) ) ).

% abs_of_pos
tff(fact_7405_abs__of__pos,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_real(abs_abs_real,A2) = A2 ) ) ).

% abs_of_pos
tff(fact_7406_abs__of__pos,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( abs_abs_rat(A2) = A2 ) ) ).

% abs_of_pos
tff(fact_7407_abs__of__pos,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_int(abs_abs_int,A2) = A2 ) ) ).

% abs_of_pos
tff(fact_7408_abs__not__less__zero,axiom,
    ! [A2: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),zero_z3403309356797280102nteger) ).

% abs_not_less_zero
tff(fact_7409_abs__not__less__zero,axiom,
    ! [A2: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A2)),zero_zero_real) ).

% abs_not_less_zero
tff(fact_7410_abs__not__less__zero,axiom,
    ! [A2: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(A2)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_7411_abs__not__less__zero,axiom,
    ! [A2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A2)),zero_zero_int) ).

% abs_not_less_zero
tff(fact_7412_abs__triangle__ineq,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B2))) ).

% abs_triangle_ineq
tff(fact_7413_abs__triangle__ineq,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2))) ).

% abs_triangle_ineq
tff(fact_7414_abs__triangle__ineq,axiom,
    ! [A2: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(plus_plus_rat(A2,B2))),plus_plus_rat(abs_abs_rat(A2),abs_abs_rat(B2))) ).

% abs_triangle_ineq
tff(fact_7415_abs__triangle__ineq,axiom,
    ! [A2: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2))) ).

% abs_triangle_ineq
tff(fact_7416_abs__mult__less,axiom,
    ! [A2: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B2)),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B2))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D)) ) ) ).

% abs_mult_less
tff(fact_7417_abs__mult__less,axiom,
    ! [A2: real,C: real,B2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A2)),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,B2)),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2))),aa_real_real(times_times_real(C),D)) ) ) ).

% abs_mult_less
tff(fact_7418_abs__mult__less,axiom,
    ! [A2: rat,C: rat,B2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(A2)),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(B2)),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(abs_abs_rat(A2)),abs_abs_rat(B2))),aa_rat_rat(times_times_rat(C),D)) ) ) ).

% abs_mult_less
tff(fact_7419_abs__mult__less,axiom,
    ! [A2: int,C: int,B2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A2)),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B2)),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D)) ) ) ).

% abs_mult_less
tff(fact_7420_abs__triangle__ineq2__sym,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B2))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B2),A2))) ).

% abs_triangle_ineq2_sym
tff(fact_7421_abs__triangle__ineq2__sym,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B2),A2))) ).

% abs_triangle_ineq2_sym
tff(fact_7422_abs__triangle__ineq2__sym,axiom,
    ! [A2: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(abs_abs_rat(A2),abs_abs_rat(B2))),abs_abs_rat(minus_minus_rat(B2,A2))) ).

% abs_triangle_ineq2_sym
tff(fact_7423_abs__triangle__ineq2__sym,axiom,
    ! [A2: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A2))) ).

% abs_triangle_ineq2_sym
tff(fact_7424_abs__triangle__ineq3,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B2)))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2))) ).

% abs_triangle_ineq3
tff(fact_7425_abs__triangle__ineq3,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2)))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A2),B2))) ).

% abs_triangle_ineq3
tff(fact_7426_abs__triangle__ineq3,axiom,
    ! [A2: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(abs_abs_rat(A2),abs_abs_rat(B2)))),abs_abs_rat(minus_minus_rat(A2,B2))) ).

% abs_triangle_ineq3
tff(fact_7427_abs__triangle__ineq3,axiom,
    ! [A2: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2))) ).

% abs_triangle_ineq3
tff(fact_7428_abs__triangle__ineq2,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B2))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2))) ).

% abs_triangle_ineq2
tff(fact_7429_abs__triangle__ineq2,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A2),B2))) ).

% abs_triangle_ineq2
tff(fact_7430_abs__triangle__ineq2,axiom,
    ! [A2: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(abs_abs_rat(A2),abs_abs_rat(B2))),abs_abs_rat(minus_minus_rat(A2,B2))) ).

% abs_triangle_ineq2
tff(fact_7431_abs__triangle__ineq2,axiom,
    ! [A2: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2))) ).

% abs_triangle_ineq2
tff(fact_7432_nonzero__abs__divide,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(A2),B2)) = aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2)) ) ) ).

% nonzero_abs_divide
tff(fact_7433_nonzero__abs__divide,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( abs_abs_rat(divide_divide_rat(A2,B2)) = divide_divide_rat(abs_abs_rat(A2),abs_abs_rat(B2)) ) ) ).

% nonzero_abs_divide
tff(fact_7434_abs__leI,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),B2) ) ) ).

% abs_leI
tff(fact_7435_abs__leI,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),B2) ) ) ).

% abs_leI
tff(fact_7436_abs__leI,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A2)),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A2)),B2) ) ) ).

% abs_leI
tff(fact_7437_abs__leI,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),B2) ) ) ).

% abs_leI
tff(fact_7438_abs__le__D2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B2) ) ).

% abs_le_D2
tff(fact_7439_abs__le__D2,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),B2) ) ).

% abs_le_D2
tff(fact_7440_abs__le__D2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A2)),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A2)),B2) ) ).

% abs_le_D2
tff(fact_7441_abs__le__D2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),B2) ) ).

% abs_le_D2
tff(fact_7442_abs__le__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),B2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B2) ) ) ).

% abs_le_iff
tff(fact_7443_abs__le__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),B2) ) ) ).

% abs_le_iff
tff(fact_7444_abs__le__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A2)),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A2)),B2) ) ) ).

% abs_le_iff
tff(fact_7445_abs__le__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),B2) ) ) ).

% abs_le_iff
tff(fact_7446_abs__ge__minus__self,axiom,
    ! [A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)) ).

% abs_ge_minus_self
tff(fact_7447_abs__ge__minus__self,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(abs_abs_real,A2)) ).

% abs_ge_minus_self
tff(fact_7448_abs__ge__minus__self,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(A2)),abs_abs_rat(A2)) ).

% abs_ge_minus_self
tff(fact_7449_abs__ge__minus__self,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),aa_int_int(abs_abs_int,A2)) ).

% abs_ge_minus_self
tff(fact_7450_abs__less__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),B2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B2)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2)),B2) ) ) ).

% abs_less_iff
tff(fact_7451_abs__less__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A2)),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A2)),B2) ) ) ).

% abs_less_iff
tff(fact_7452_abs__less__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(A2)),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,uminus_uminus_rat(A2)),B2) ) ) ).

% abs_less_iff
tff(fact_7453_abs__less__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A2)),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A2)),B2) ) ) ).

% abs_less_iff
tff(fact_7454_sin__x__le__x,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),Xb) ) ).

% sin_x_le_x
tff(fact_7455_dbl__def,axiom,
    ! [Xb: real] : aa_real_real(neg_numeral_dbl_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb) ).

% dbl_def
tff(fact_7456_dbl__def,axiom,
    ! [Xb: rat] : neg_numeral_dbl_rat(Xb) = plus_plus_rat(Xb,Xb) ).

% dbl_def
tff(fact_7457_dbl__def,axiom,
    ! [Xb: int] : aa_int_int(neg_numeral_dbl_int,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb) ).

% dbl_def
tff(fact_7458_cos__squared__eq,axiom,
    ! [Xb: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_7459_cos__squared__eq,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_7460_sin__squared__eq,axiom,
    ! [Xb: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_7461_sin__squared__eq,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_7462_dense__eq0__I,axiom,
    ! [Xb: real] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),E) )
     => ( Xb = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_7463_dense__eq0__I,axiom,
    ! [Xb: rat] :
      ( ! [E: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(Xb)),E) )
     => ( Xb = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_7464_abs__mult__pos,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y)),Xb) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_7465_abs__mult__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(times_times_real(aa_real_real(abs_abs_real,Y)),Xb) = aa_real_real(abs_abs_real,aa_real_real(times_times_real(Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_7466_abs__mult__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_rat(times_times_rat(abs_abs_rat(Y)),Xb) = abs_abs_rat(aa_rat_rat(times_times_rat(Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_7467_abs__mult__pos,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),Xb) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_7468_abs__eq__mult,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),zero_z3403309356797280102nteger) ) )
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B2)) ) ) ).

% abs_eq_mult
tff(fact_7469_abs__eq__mult,axiom,
    ! [A2: real,B2: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(times_times_real(A2),B2)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2)) ) ) ).

% abs_eq_mult
tff(fact_7470_abs__eq__mult,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) ) )
     => ( abs_abs_rat(aa_rat_rat(times_times_rat(A2),B2)) = aa_rat_rat(times_times_rat(abs_abs_rat(A2)),abs_abs_rat(B2)) ) ) ).

% abs_eq_mult
tff(fact_7471_abs__eq__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2)) ) ) ).

% abs_eq_mult
tff(fact_7472_abs__minus__le__zero,axiom,
    ! [A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2))),zero_z3403309356797280102nteger) ).

% abs_minus_le_zero
tff(fact_7473_abs__minus__le__zero,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A2))),zero_zero_real) ).

% abs_minus_le_zero
tff(fact_7474_abs__minus__le__zero,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,uminus_uminus_rat(abs_abs_rat(A2))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_7475_abs__minus__le__zero,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A2))),zero_zero_int) ).

% abs_minus_le_zero
tff(fact_7476_eq__abs__iff_H,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B2) )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
        & ( ( B2 = A2 )
          | ( B2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7477_eq__abs__iff_H,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = aa_real_real(abs_abs_real,B2) )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
        & ( ( B2 = A2 )
          | ( B2 = aa_real_real(uminus_uminus_real,A2) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7478_eq__abs__iff_H,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = abs_abs_rat(B2) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
        & ( ( B2 = A2 )
          | ( B2 = uminus_uminus_rat(A2) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7479_eq__abs__iff_H,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(abs_abs_int,B2) )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
        & ( ( B2 = A2 )
          | ( B2 = aa_int_int(uminus_uminus_int,A2) ) ) ) ) ).

% eq_abs_iff'
tff(fact_7480_abs__eq__iff_H,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2) = B2 )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
        & ( ( A2 = B2 )
          | ( A2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7481_abs__eq__iff_H,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(abs_abs_real,A2) = B2 )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
        & ( ( A2 = B2 )
          | ( A2 = aa_real_real(uminus_uminus_real,B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7482_abs__eq__iff_H,axiom,
    ! [A2: rat,B2: rat] :
      ( ( abs_abs_rat(A2) = B2 )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
        & ( ( A2 = B2 )
          | ( A2 = uminus_uminus_rat(B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7483_abs__eq__iff_H,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(abs_abs_int,A2) = B2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
        & ( ( A2 = B2 )
          | ( A2 = aa_int_int(uminus_uminus_int,B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_7484_abs__div__pos,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,Xb)),Y) = aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(Xb),Y)) ) ) ).

% abs_div_pos
tff(fact_7485_abs__div__pos,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( divide_divide_rat(abs_abs_rat(Xb),Y) = abs_abs_rat(divide_divide_rat(Xb,Y)) ) ) ).

% abs_div_pos
tff(fact_7486_zero__le__power__abs,axiom,
    ! [A2: real,N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),N)) ).

% zero_le_power_abs
tff(fact_7487_zero__le__power__abs,axiom,
    ! [A2: code_integer,N: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),N)) ).

% zero_le_power_abs
tff(fact_7488_zero__le__power__abs,axiom,
    ! [A2: rat,N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(abs_abs_rat(A2)),N)) ).

% zero_le_power_abs
tff(fact_7489_zero__le__power__abs,axiom,
    ! [A2: int,N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A2)),N)) ).

% zero_le_power_abs
tff(fact_7490_abs__if,axiom,
    ! [A2: code_integer] :
      aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2),A2) ).

% abs_if
tff(fact_7491_abs__if,axiom,
    ! [A2: real] :
      aa_real_real(abs_abs_real,A2) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real),aa_real_real(uminus_uminus_real,A2),A2) ).

% abs_if
tff(fact_7492_abs__if,axiom,
    ! [A2: rat] :
      abs_abs_rat(A2) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat),uminus_uminus_rat(A2),A2) ).

% abs_if
tff(fact_7493_abs__if,axiom,
    ! [A2: int] :
      aa_int_int(abs_abs_int,A2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int),aa_int_int(uminus_uminus_int,A2),A2) ).

% abs_if
tff(fact_7494_abs__of__neg,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger)
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) ) ) ).

% abs_of_neg
tff(fact_7495_abs__of__neg,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A2) = aa_real_real(uminus_uminus_real,A2) ) ) ).

% abs_of_neg
tff(fact_7496_abs__of__neg,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( abs_abs_rat(A2) = uminus_uminus_rat(A2) ) ) ).

% abs_of_neg
tff(fact_7497_abs__of__neg,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A2) = aa_int_int(uminus_uminus_int,A2) ) ) ).

% abs_of_neg
tff(fact_7498_abs__if__raw,axiom,
    ! [X3: code_integer] :
      aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X3) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X3),X3) ).

% abs_if_raw
tff(fact_7499_abs__if__raw,axiom,
    ! [X3: real] :
      aa_real_real(abs_abs_real,X3) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X3),zero_zero_real),aa_real_real(uminus_uminus_real,X3),X3) ).

% abs_if_raw
tff(fact_7500_abs__if__raw,axiom,
    ! [X3: rat] :
      abs_abs_rat(X3) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),zero_zero_rat),uminus_uminus_rat(X3),X3) ).

% abs_if_raw
tff(fact_7501_abs__if__raw,axiom,
    ! [X3: int] :
      aa_int_int(abs_abs_int,X3) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X3),zero_zero_int),aa_int_int(uminus_uminus_int,X3),X3) ).

% abs_if_raw
tff(fact_7502_abs__diff__le__iff,axiom,
    ! [Xb: code_integer,A2: code_integer,R: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Xb),A2))),R)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),R)),Xb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),R)) ) ) ).

% abs_diff_le_iff
tff(fact_7503_abs__diff__le__iff,axiom,
    ! [Xb: real,A2: real,R: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),A2))),R)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A2),R)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),R)) ) ) ).

% abs_diff_le_iff
tff(fact_7504_abs__diff__le__iff,axiom,
    ! [Xb: rat,A2: rat,R: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(Xb,A2))),R)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,minus_minus_rat(A2,R)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),plus_plus_rat(A2,R)) ) ) ).

% abs_diff_le_iff
tff(fact_7505_abs__diff__le__iff,axiom,
    ! [Xb: int,A2: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A2))),R)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),R)),Xb)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),R)) ) ) ).

% abs_diff_le_iff
tff(fact_7506_abs__diff__triangle__ineq,axiom,
    ! [A2: code_integer,B2: code_integer,C: code_integer,D: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),B2)),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(C),D)))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),C))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(B2),D)))) ).

% abs_diff_triangle_ineq
tff(fact_7507_abs__diff__triangle__ineq,axiom,
    ! [A2: real,B2: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A2),C))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B2),D)))) ).

% abs_diff_triangle_ineq
tff(fact_7508_abs__diff__triangle__ineq,axiom,
    ! [A2: rat,B2: rat,C: rat,D: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(plus_plus_rat(A2,B2),plus_plus_rat(C,D)))),plus_plus_rat(abs_abs_rat(minus_minus_rat(A2,C)),abs_abs_rat(minus_minus_rat(B2,D)))) ).

% abs_diff_triangle_ineq
tff(fact_7509_abs__diff__triangle__ineq,axiom,
    ! [A2: int,B2: int,C: int,D: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),D)))) ).

% abs_diff_triangle_ineq
tff(fact_7510_abs__triangle__ineq4,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),B2))),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B2))) ).

% abs_triangle_ineq4
tff(fact_7511_abs__triangle__ineq4,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A2),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2))) ).

% abs_triangle_ineq4
tff(fact_7512_abs__triangle__ineq4,axiom,
    ! [A2: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(A2,B2))),plus_plus_rat(abs_abs_rat(A2),abs_abs_rat(B2))) ).

% abs_triangle_ineq4
tff(fact_7513_abs__triangle__ineq4,axiom,
    ! [A2: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2))) ).

% abs_triangle_ineq4
tff(fact_7514_abs__diff__less__iff,axiom,
    ! [Xb: code_integer,A2: code_integer,R: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Xb),A2))),R)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(A2),R)),Xb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(A2),R)) ) ) ).

% abs_diff_less_iff
tff(fact_7515_abs__diff__less__iff,axiom,
    ! [Xb: real,A2: real,R: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),A2))),R)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A2),R)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),R)) ) ) ).

% abs_diff_less_iff
tff(fact_7516_abs__diff__less__iff,axiom,
    ! [Xb: rat,A2: rat,R: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(minus_minus_rat(Xb,A2))),R)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,minus_minus_rat(A2,R)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),plus_plus_rat(A2,R)) ) ) ).

% abs_diff_less_iff
tff(fact_7517_abs__diff__less__iff,axiom,
    ! [Xb: int,A2: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A2))),R)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),R)),Xb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),R)) ) ) ).

% abs_diff_less_iff
tff(fact_7518_sin__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_gt_zero
tff(fact_7519_sin__x__ge__neg__x,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(sin_real,Xb)) ) ).

% sin_x_ge_neg_x
tff(fact_7520_sin__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_ge_zero
tff(fact_7521_cos__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( ( aa_real_real(cos_real,Xb) = aa_real_real(cos_real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_7522_cos__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_7523_cos__monotone__0__pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_7524_cos__diff__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(minus_minus_real(aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one)))))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Z),W)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_7525_sin__diff__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(minus_minus_real(aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W),Z)),numeral_numeral_real(aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_7526_sin__plus__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W),Z)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_7527_cos__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(times_times_real(aa_real_real(cos_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z))),aa_real_real(sin_real,aa_real_real(minus_minus_real(W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_7528_sin__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(times_times_real(aa_real_real(sin_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z))),aa_real_real(sin_real,aa_real_real(minus_minus_real(W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_7529_sin__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(times_times_real(aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(cos_real,aa_real_real(minus_minus_real(W),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_7530_abs__real__def,axiom,
    ! [A2: real] :
      aa_real_real(abs_abs_real,A2) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real),aa_real_real(uminus_uminus_real,A2),A2) ).

% abs_real_def
tff(fact_7531_cos__double,axiom,
    ! [Xb: complex] : aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_7532_cos__double,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),Xb)) = aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_7533_lemma__interval__lt,axiom,
    ! [A2: real,Xb: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B2)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [Y4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),Y4))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),B2) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_7534_cos__double__sin,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_7535_cos__double__sin,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),W)) = aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_7536_abs__add__one__gt__zero,axiom,
    ! [Xb: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb))) ).

% abs_add_one_gt_zero
tff(fact_7537_abs__add__one__gt__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,Xb))) ).

% abs_add_one_gt_zero
tff(fact_7538_abs__add__one__gt__zero,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),plus_plus_rat(one_one_rat,abs_abs_rat(Xb))) ).

% abs_add_one_gt_zero
tff(fact_7539_abs__add__one__gt__zero,axiom,
    ! [Xb: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,Xb))) ).

% abs_add_one_gt_zero
tff(fact_7540_of__int__leD,axiom,
    ! [N: int,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(ring_18347121197199848620nteger,N))),Xb)
     => ( ( N = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),Xb) ) ) ).

% of_int_leD
tff(fact_7541_of__int__leD,axiom,
    ! [N: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N))),Xb)
     => ( ( N = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% of_int_leD
tff(fact_7542_of__int__leD,axiom,
    ! [N: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(aa_int_rat(ring_1_of_int_rat,N))),Xb)
     => ( ( N = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ) ).

% of_int_leD
tff(fact_7543_of__int__leD,axiom,
    ! [N: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),Xb)
     => ( ( N = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Xb) ) ) ).

% of_int_leD
tff(fact_7544_of__int__lessD,axiom,
    ! [N: int,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(ring_18347121197199848620nteger,N))),Xb)
     => ( ( N = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),Xb) ) ) ).

% of_int_lessD
tff(fact_7545_of__int__lessD,axiom,
    ! [N: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,N))),Xb)
     => ( ( N = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% of_int_lessD
tff(fact_7546_of__int__lessD,axiom,
    ! [N: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(aa_int_rat(ring_1_of_int_rat,N))),Xb)
     => ( ( N = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ) ).

% of_int_lessD
tff(fact_7547_of__int__lessD,axiom,
    ! [N: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,N))),Xb)
     => ( ( N = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Xb) ) ) ).

% of_int_lessD
tff(fact_7548_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_7549_cos__monotone__0__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_7550_cos__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_7551_sin__eq__0__pi,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),pi)
       => ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
         => ( Xb = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_7552_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_7553_lemma__interval,axiom,
    ! [A2: real,Xb: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B2)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [Y4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),Y4))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),B2) ) ) ) ) ) ).

% lemma_interval
tff(fact_7554_round__diff__minimal,axiom,
    ! [Z: real,M: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Z))))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z),aa_int_real(ring_1_of_int_real,M)))) ).

% round_diff_minimal
tff(fact_7555_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(Z,aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Z))))),abs_abs_rat(minus_minus_rat(Z,aa_int_rat(ring_1_of_int_rat,M)))) ).

% round_diff_minimal
tff(fact_7556_sin__zero__iff__int2,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ? [I4: int] : Xb = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I4)),pi) ) ).

% sin_zero_iff_int2
tff(fact_7557_sincos__total__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),pi)
            & ( Xb = aa_real_real(cos_real,T) )
            & ( Y = aa_real_real(sin_real,T) ) ) ) ) ).

% sincos_total_pi
tff(fact_7558_sin__expansion__lemma,axiom,
    ! [Xb: real,M: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_7559_abs__le__square__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_7560_abs__le__square__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_7561_abs__le__square__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(Xb)),abs_abs_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_7562_abs__le__square__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),aa_int_int(abs_abs_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_7563_abs__square__eq__1,axiom,
    ! [Xb: rat] :
      ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( abs_abs_rat(Xb) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_7564_abs__square__eq__1,axiom,
    ! [Xb: int] :
      ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,Xb) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_7565_abs__square__eq__1,axiom,
    ! [Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( aa_real_real(abs_abs_real,Xb) = one_one_real ) ) ).

% abs_square_eq_1
tff(fact_7566_abs__square__eq__1,axiom,
    ! [Xb: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_7567_power__even__abs,axiom,
    ! [N: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_rat(power_power_rat(abs_abs_rat(A2)),N) = aa_nat_rat(power_power_rat(A2),N) ) ) ).

% power_even_abs
tff(fact_7568_power__even__abs,axiom,
    ! [N: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A2)),N) = aa_nat_int(power_power_int(A2),N) ) ) ).

% power_even_abs
tff(fact_7569_power__even__abs,axiom,
    ! [N: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N) ) ) ).

% power_even_abs
tff(fact_7570_power__even__abs,axiom,
    ! [N: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),N) = aa_nat_Code_integer(power_8256067586552552935nteger(A2),N) ) ) ).

% power_even_abs
tff(fact_7571_cos__expansion__lemma,axiom,
    ! [Xb: real,M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),numeral_numeral_real(aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M)),pi)),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_7572_sin__gt__zero__02,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),numeral_numeral_real(aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_gt_zero_02
tff(fact_7573_cos__two__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_less_zero
tff(fact_7574_cos__two__le__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,numeral_numeral_real(aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_le_zero
tff(fact_7575_cos__is__zero,axiom,
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),numeral_numeral_real(aa_num_num(bit0,one)))
      & ( aa_real_real(cos_real,X2) = zero_zero_real )
      & ! [Y4: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),numeral_numeral_real(aa_num_num(bit0,one)))
            & ( aa_real_real(cos_real,Y4) = zero_zero_real ) )
         => ( Y4 = X2 ) ) ) ).

% cos_is_zero
tff(fact_7576_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% signed_take_bit_int_less_exp
tff(fact_7577_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_7578_cos__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),pi)
            & ( aa_real_real(cos_real,X2) = Y )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),pi)
                  & ( aa_real_real(cos_real,Y4) = Y ) )
               => ( Y4 = X2 ) ) ) ) ) ).

% cos_total
tff(fact_7579_even__signed__take__bit__iff,axiom,
    ! [M: nat,A2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),bit_ri6224792872505173163uint32(M,A2))
    <=> aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),A2) ) ).

% even_signed_take_bit_iff
tff(fact_7580_even__signed__take__bit__iff,axiom,
    ! [M: nat,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(M),A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2) ) ).

% even_signed_take_bit_iff
tff(fact_7581_even__signed__take__bit__iff,axiom,
    ! [M: nat,A2: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),bit_ri1375673916561920181l_num1(M,A2))
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),A2) ) ).

% even_signed_take_bit_iff
tff(fact_7582_even__signed__take__bit__iff,axiom,
    ! [M: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(M),A2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2) ) ).

% even_signed_take_bit_iff
tff(fact_7583_sincos__total__pi__half,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
              & ( Xb = aa_real_real(cos_real,T) )
              & ( Y = aa_real_real(sin_real,T) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_7584_sincos__total__2pi__le,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))
          & ( Xb = aa_real_real(cos_real,T) )
          & ( Y = aa_real_real(sin_real,T) ) ) ) ).

% sincos_total_2pi_le
tff(fact_7585_power2__le__iff__abs__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7586_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7587_power2__le__iff__abs__le,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7588_power2__le__iff__abs__le,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_7589_abs__square__le__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real) ) ).

% abs_square_le_1
tff(fact_7590_abs__square__le__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_7591_abs__square__le__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(Xb)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_7592_abs__square__le__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),one_one_int) ) ).

% abs_square_le_1
tff(fact_7593_abs__square__less__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_7594_abs__square__less__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real) ) ).

% abs_square_less_1
tff(fact_7595_abs__square__less__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(Xb)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_7596_abs__square__less__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Xb)),one_one_int) ) ).

% abs_square_less_1
tff(fact_7597_power__mono__even,axiom,
    ! [N: nat,A2: real,B2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),N)) ) ) ).

% power_mono_even
tff(fact_7598_power__mono__even,axiom,
    ! [N: nat,A2: code_integer,B2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,A2)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,B2))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A2),N)),aa_nat_Code_integer(power_8256067586552552935nteger(B2),N)) ) ) ).

% power_mono_even
tff(fact_7599_power__mono__even,axiom,
    ! [N: nat,A2: rat,B2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(A2)),abs_abs_rat(B2))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N)),aa_nat_rat(power_power_rat(B2),N)) ) ) ).

% power_mono_even
tff(fact_7600_power__mono__even,axiom,
    ! [N: nat,A2: int,B2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),N)),aa_nat_int(power_power_int(B2),N)) ) ) ).

% power_mono_even
tff(fact_7601_sincos__total__2pi,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))
             => ( ( Xb = aa_real_real(cos_real,T) )
               => ( Y != aa_real_real(sin_real,T) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_7602_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_ri631733984087533419it_int(N),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_7603_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_7604_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_int_int(bit_ri631733984087533419it_int(N),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_7605_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_7606_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_ri631733984087533419it_int(N),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_7607_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,N)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_7608_cos__plus__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)),numeral_numeral_real(aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W),Z)),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_7609_cos__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(times_times_real(aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(minus_minus_real(W),Z))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Z)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_7610_sin__gt__zero2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_gt_zero2
tff(fact_7611_sin__lt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_7612_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_30
tff(fact_7613_cos__double__less__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),numeral_numeral_real(aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),Xb))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_7614_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_7615_cos__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_gt_zero
tff(fact_7616_sin__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( ( aa_real_real(sin_real,Xb) = aa_real_real(sin_real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_7617_sin__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_7618_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,Xb)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_7619_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_60
tff(fact_7620_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(N),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_7621_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_int_int(bit_ri631733984087533419it_int(N),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_7622_cos__one__2pi__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
    <=> ? [X: int] : Xb = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,X)),numeral_numeral_real(aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_7623_cos__double__cos,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_complex) ).

% cos_double_cos
tff(fact_7624_cos__double__cos,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),W)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_real) ).

% cos_double_cos
tff(fact_7625_cos__treble__cos,axiom,
    ! [Xb: complex] : aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit1,one))),Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit1,one))),aa_complex_complex(cos_complex,Xb))) ).

% cos_treble_cos
tff(fact_7626_cos__treble__cos,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit1,one))),Xb)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit1,one))),aa_real_real(cos_real,Xb))) ).

% cos_treble_cos
tff(fact_7627_sin__le__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_7628_sin__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_7629_sin__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_7630_sin__monotone__2pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,Xb)) ) ) ) ).

% sin_monotone_2pi
tff(fact_7631_sin__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),X2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
            & ( aa_real_real(sin_real,X2) = Y )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
                  & ( aa_real_real(sin_real,Y4) = Y ) )
               => ( Y4 = X2 ) ) ) ) ) ).

% sin_total
tff(fact_7632_cos__gt__zero__pi,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_gt_zero_pi
tff(fact_7633_cos__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_ge_zero
tff(fact_7634_cos__one__2pi,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
    <=> ( ? [X: nat] : Xb = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X)),numeral_numeral_real(aa_num_num(bit0,one)))),pi)
        | ? [X: nat] : Xb = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X)),numeral_numeral_real(aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_7635_of__int__round__abs__le,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))),Xb))),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_7636_of__int__round__abs__le,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,abs_abs_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,archim7778729529865785530nd_rat(Xb)),Xb))),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_7637_round__unique_H,axiom,
    ! [Xb: real,N: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),aa_int_real(ring_1_of_int_real,N)))),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
     => ( aa_real_int(archim8280529875227126926d_real,Xb) = N ) ) ).

% round_unique'
tff(fact_7638_round__unique_H,axiom,
    ! [Xb: rat,N: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,abs_abs_rat(minus_minus_rat(Xb,aa_int_rat(ring_1_of_int_rat,N)))),divide_divide_rat(one_one_rat,numeral_numeral_rat(aa_num_num(bit0,one))))
     => ( archim7778729529865785530nd_rat(Xb) = N ) ) ).

% round_unique'
tff(fact_7639_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,N)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_7640_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N)))),aa_int_int(bit_ri631733984087533419it_int(N),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_7641_signed__take__bit__Suc,axiom,
    ! [N: nat,A2: word_N3645301735248828278l_num1] : bit_ri1375673916561920181l_num1(aa_nat_nat(suc,N),A2) = plus_p361126936061061375l_num1(modulo1504961113040953224l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),bit_ri1375673916561920181l_num1(N,divide1791077408188789448l_num1(A2,numera7442385471795722001l_num1(aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_7642_signed__take__bit__Suc,axiom,
    ! [N: nat,A2: code_integer] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(aa_nat_nat(suc,N)),A2) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(N),aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(A2),numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_7643_signed__take__bit__Suc,axiom,
    ! [N: nat,A2: uint32] : bit_ri6224792872505173163uint32(aa_nat_nat(suc,N),A2) = plus_plus_uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),bit_ri6224792872505173163uint32(N,aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,A2),numera9087168376688890119uint32(aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_7644_signed__take__bit__Suc,axiom,
    ! [N: nat,A2: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_7645_sin__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ? [I4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I4)
          & ( Xb = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I4)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_7646_cos__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
    <=> ? [I4: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I4)
          & ( Xb = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I4)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_7647_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_7648_sin__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
       => ? [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)
            & ( Xb = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_7649_sin__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ( ? [N8: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8)
            & ( Xb = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N8)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) )
        | ? [N8: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8)
            & ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N8)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_7650_cos__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
       => ? [N3: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3)
            & ( Xb = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N3)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_7651_cos__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
    <=> ( ? [N8: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8)
            & ( Xb = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N8)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) )
        | ? [N8: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8)
            & ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N8)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_7652_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_7653_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,Xb: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
         => aa_real_o(aa_real_fun_real_o(P,X2),aa_nat_real(aa_real_fun_nat_real(power_power_real,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_real_o(aa_real_fun_real_o(P,aa_real_real(abs_abs_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7654_abs__sqrt__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,Xb: code_integer] :
      ( ! [X2: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X2)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,X2),aa_nat_Code_integer(power_8256067586552552935nteger(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7655_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,Xb: rat] :
      ( ! [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X2)
         => aa_rat_o(aa_rat_fun_rat_o(P,X2),aa_nat_rat(power_power_rat(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_rat_o(aa_rat_fun_rat_o(P,abs_abs_rat(Xb)),aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7656_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,Xb: int] :
      ( ! [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X2)
         => aa_int_o(aa_int_fun_int_o(P,X2),aa_nat_int(power_power_int(X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_int_o(aa_int_fun_int_o(P,aa_int_int(abs_abs_int,Xb)),aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_7657_monoseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_uk,Xb)) ) ).

% monoseq_arctan_series
tff(fact_7658_arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(arctan,Xb) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ul,Xb)) ) ) ).

% arctan_series
tff(fact_7659_summable__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_ul,Xb)) ) ).

% summable_arctan_series
tff(fact_7660_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_7661_zdvd1__eq,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),one_one_int)
    <=> ( aa_int_int(abs_abs_int,Xb) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_7662_arctan__zero__zero,axiom,
    aa_real_real(arctan,zero_zero_real) = zero_zero_real ).

% arctan_zero_zero
tff(fact_7663_arctan__eq__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(arctan,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% arctan_eq_zero_iff
tff(fact_7664_summable__zero,axiom,
    summable_complex(aTP_Lamm_um) ).

% summable_zero
tff(fact_7665_summable__zero,axiom,
    summable_real(aTP_Lamm_tz) ).

% summable_zero
tff(fact_7666_summable__zero,axiom,
    summable_nat(aTP_Lamm_iv) ).

% summable_zero
tff(fact_7667_summable__zero,axiom,
    summable_int(aTP_Lamm_ua) ).

% summable_zero
tff(fact_7668_summable__single,axiom,
    ! [I: nat,Fa: fun_nat_complex] : summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_un(I),Fa)) ).

% summable_single
tff(fact_7669_summable__single,axiom,
    ! [I: nat,Fa: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_uo(I),Fa)) ).

% summable_single
tff(fact_7670_summable__single,axiom,
    ! [I: nat,Fa: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_up(I),Fa)) ).

% summable_single
tff(fact_7671_summable__single,axiom,
    ! [I: nat,Fa: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_uq(I),Fa)) ).

% summable_single
tff(fact_7672_summable__iff__shift,axiom,
    ! [Fa: fun_nat_real,K: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_ur(Fa),K))
    <=> summable_real(Fa) ) ).

% summable_iff_shift
tff(fact_7673_summable__iff__shift,axiom,
    ! [Fa: fun_nat_complex,K: nat] :
      ( summable_complex(aa_nat6454746962644518025omplex(aTP_Lamm_us(Fa),K))
    <=> summable_complex(Fa) ) ).

% summable_iff_shift
tff(fact_7674_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int)
    <=> ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_7675_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_7676_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( aa_nat_nat(suc,N) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( N = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_7677_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,N) )
    <=> ( aa_num_nat(pred_numeral,K) = N ) ) ).

% eq_numeral_Suc
tff(fact_7678_arctan__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% arctan_less_zero_iff
tff(fact_7679_zero__less__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arctan,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% zero_less_arctan_iff
tff(fact_7680_arctan__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% arctan_le_zero_iff
tff(fact_7681_zero__le__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% zero_le_arctan_iff
tff(fact_7682_summable__cmult__iff,axiom,
    ! [C: complex,Fa: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ut(C),Fa))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(Fa) ) ) ).

% summable_cmult_iff
tff(fact_7683_summable__cmult__iff,axiom,
    ! [C: real,Fa: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_uu(C),Fa))
    <=> ( ( C = zero_zero_real )
        | summable_real(Fa) ) ) ).

% summable_cmult_iff
tff(fact_7684_summable__divide__iff,axiom,
    ! [Fa: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_uv(Fa),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(Fa) ) ) ).

% summable_divide_iff
tff(fact_7685_summable__divide__iff,axiom,
    ! [Fa: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_uw(Fa),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(Fa) ) ) ).

% summable_divide_iff
tff(fact_7686_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit1,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K)) ).

% pred_numeral_simps(3)
tff(fact_7687_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K)),N) ) ).

% less_numeral_Suc
tff(fact_7688_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_num_nat(pred_numeral,K)) ) ).

% less_Suc_numeral
tff(fact_7689_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),N) ) ).

% le_numeral_Suc
tff(fact_7690_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_num_nat(pred_numeral,K)) ) ).

% le_Suc_numeral
tff(fact_7691_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_7692_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),N) ).

% diff_numeral_Suc
tff(fact_7693_max__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,N),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_7694_max__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),N)) ).

% max_numeral_Suc
tff(fact_7695_summable__geometric__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(power_power_real,C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real) ) ).

% summable_geometric_iff
tff(fact_7696_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(power_power_complex,C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real) ) ).

% summable_geometric_iff
tff(fact_7697_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_bit0
tff(fact_7698_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_7699_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_7700_summable__norm__cancel,axiom,
    ! [Fa: fun_nat_real] :
      ( summable_real(aTP_Lamm_ux(Fa))
     => summable_real(Fa) ) ).

% summable_norm_cancel
tff(fact_7701_summable__norm__cancel,axiom,
    ! [Fa: fun_nat_complex] :
      ( summable_real(aTP_Lamm_uy(Fa))
     => summable_complex(Fa) ) ).

% summable_norm_cancel
tff(fact_7702_arctan__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% arctan_less_iff
tff(fact_7703_arctan__monotone,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y)) ) ).

% arctan_monotone
tff(fact_7704_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N2: nat,Fa: fun_nat_real] :
      ( summable_real(G)
     => ( ! [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Fa,N3))),aa_nat_real(G,N3)) )
       => summable_real(Fa) ) ) ).

% summable_comparison_test'
tff(fact_7705_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N2: nat,Fa: fun_nat_complex] :
      ( summable_real(G)
     => ( ! [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Fa,N3))),aa_nat_real(G,N3)) )
       => summable_complex(Fa) ) ) ).

% summable_comparison_test'
tff(fact_7706_summable__comparison__test,axiom,
    ! [Fa: fun_nat_real,G: fun_nat_real] :
      ( ? [N9: nat] :
        ! [N3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N9),N3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Fa,N3))),aa_nat_real(G,N3)) )
     => ( summable_real(G)
       => summable_real(Fa) ) ) ).

% summable_comparison_test
tff(fact_7707_summable__comparison__test,axiom,
    ! [Fa: fun_nat_complex,G: fun_nat_real] :
      ( ? [N9: nat] :
        ! [N3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N9),N3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Fa,N3))),aa_nat_real(G,N3)) )
     => ( summable_real(G)
       => summable_complex(Fa) ) ) ).

% summable_comparison_test
tff(fact_7708_summable__const__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_uz,C))
    <=> ( C = zero_zero_complex ) ) ).

% summable_const_iff
tff(fact_7709_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_va,C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_7710_summable__add,axiom,
    ! [Fa: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(Fa)
     => ( summable_complex(G)
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vb(Fa),G)) ) ) ).

% summable_add
tff(fact_7711_summable__add,axiom,
    ! [Fa: fun_nat_real,G: fun_nat_real] :
      ( summable_real(Fa)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_vc(Fa),G)) ) ) ).

% summable_add
tff(fact_7712_summable__add,axiom,
    ! [Fa: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(Fa)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_vd(Fa),G)) ) ) ).

% summable_add
tff(fact_7713_summable__add,axiom,
    ! [Fa: fun_nat_int,G: fun_nat_int] :
      ( summable_int(Fa)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ve(Fa),G)) ) ) ).

% summable_add
tff(fact_7714_summable__mult2,axiom,
    ! [Fa: fun_nat_complex,C: complex] :
      ( summable_complex(Fa)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_vf(Fa),C)) ) ).

% summable_mult2
tff(fact_7715_summable__mult2,axiom,
    ! [Fa: fun_nat_real,C: real] :
      ( summable_real(Fa)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_vg(Fa),C)) ) ).

% summable_mult2
tff(fact_7716_summable__mult,axiom,
    ! [Fa: fun_nat_complex,C: complex] :
      ( summable_complex(Fa)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_vh(Fa),C)) ) ).

% summable_mult
tff(fact_7717_summable__mult,axiom,
    ! [Fa: fun_nat_real,C: real] :
      ( summable_real(Fa)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_vi(Fa),C)) ) ).

% summable_mult
tff(fact_7718_summable__diff,axiom,
    ! [Fa: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(Fa)
     => ( summable_complex(G)
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vj(Fa),G)) ) ) ).

% summable_diff
tff(fact_7719_summable__diff,axiom,
    ! [Fa: fun_nat_real,G: fun_nat_real] :
      ( summable_real(Fa)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_vk(Fa),G)) ) ) ).

% summable_diff
tff(fact_7720_summable__Suc__iff,axiom,
    ! [Fa: fun_nat_real] :
      ( summable_real(aTP_Lamm_vl(Fa))
    <=> summable_real(Fa) ) ).

% summable_Suc_iff
tff(fact_7721_summable__Suc__iff,axiom,
    ! [Fa: fun_nat_complex] :
      ( summable_complex(aTP_Lamm_vm(Fa))
    <=> summable_complex(Fa) ) ).

% summable_Suc_iff
tff(fact_7722_summable__divide,axiom,
    ! [Fa: fun_nat_complex,C: complex] :
      ( summable_complex(Fa)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_uv(Fa),C)) ) ).

% summable_divide
tff(fact_7723_summable__divide,axiom,
    ! [Fa: fun_nat_real,C: real] :
      ( summable_real(Fa)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_uw(Fa),C)) ) ).

% summable_divide
tff(fact_7724_summable__minus__iff,axiom,
    ! [Fa: fun_nat_complex] :
      ( summable_complex(aTP_Lamm_vn(Fa))
    <=> summable_complex(Fa) ) ).

% summable_minus_iff
tff(fact_7725_summable__minus__iff,axiom,
    ! [Fa: fun_nat_real] :
      ( summable_real(aTP_Lamm_vo(Fa))
    <=> summable_real(Fa) ) ).

% summable_minus_iff
tff(fact_7726_summable__minus,axiom,
    ! [Fa: fun_nat_complex] :
      ( summable_complex(Fa)
     => summable_complex(aTP_Lamm_vn(Fa)) ) ).

% summable_minus
tff(fact_7727_summable__minus,axiom,
    ! [Fa: fun_nat_real] :
      ( summable_real(Fa)
     => summable_real(aTP_Lamm_vo(Fa)) ) ).

% summable_minus
tff(fact_7728_summable__ignore__initial__segment,axiom,
    ! [Fa: fun_nat_real,K: nat] :
      ( summable_real(Fa)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_ur(Fa),K)) ) ).

% summable_ignore_initial_segment
tff(fact_7729_summable__ignore__initial__segment,axiom,
    ! [Fa: fun_nat_complex,K: nat] :
      ( summable_complex(Fa)
     => summable_complex(aa_nat6454746962644518025omplex(aTP_Lamm_us(Fa),K)) ) ).

% summable_ignore_initial_segment
tff(fact_7730_summable__rabs__cancel,axiom,
    ! [Fa: fun_nat_real] :
      ( summable_real(aTP_Lamm_vp(Fa))
     => summable_real(Fa) ) ).

% summable_rabs_cancel
tff(fact_7731_powser__insidea,axiom,
    ! [Fa: fun_nat_real,Xb: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_vq(Fa),Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,Xb))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_vr(Fa),Z)) ) ) ).

% powser_insidea
tff(fact_7732_powser__insidea,axiom,
    ! [Fa: fun_nat_complex,Xb: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_vs(Fa),Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,Xb))
       => summable_real(aa_com20334259897932645t_real(aTP_Lamm_vt(Fa),Z)) ) ) ).

% powser_insidea
tff(fact_7733_suminf__le,axiom,
    ! [Fa: fun_nat_real,G: fun_nat_real] :
      ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Fa,N3)),aa_nat_real(G,N3))
     => ( summable_real(Fa)
       => ( summable_real(G)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(Fa)),suminf_real(G)) ) ) ) ).

% suminf_le
tff(fact_7734_suminf__le,axiom,
    ! [Fa: fun_nat_nat,G: fun_nat_nat] :
      ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Fa,N3)),aa_nat_nat(G,N3))
     => ( summable_nat(Fa)
       => ( summable_nat(G)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(Fa)),suminf_nat(G)) ) ) ) ).

% suminf_le
tff(fact_7735_suminf__le,axiom,
    ! [Fa: fun_nat_int,G: fun_nat_int] :
      ( ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,N3)),aa_nat_int(G,N3))
     => ( summable_int(Fa)
       => ( summable_int(G)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(Fa)),suminf_int(G)) ) ) ) ).

% suminf_le
tff(fact_7736_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),N)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_7737_summable__mult__D,axiom,
    ! [C: complex,Fa: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ut(C),Fa))
     => ( ( C != zero_zero_complex )
       => summable_complex(Fa) ) ) ).

% summable_mult_D
tff(fact_7738_summable__mult__D,axiom,
    ! [C: real,Fa: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_uu(C),Fa))
     => ( ( C != zero_zero_real )
       => summable_real(Fa) ) ) ).

% summable_mult_D
tff(fact_7739_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_7740_summable__zero__power,axiom,
    summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).

% summable_zero_power
tff(fact_7741_summable__zero__power,axiom,
    summable_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex)) ).

% summable_zero_power
tff(fact_7742_numeral__eq__Suc,axiom,
    ! [K: num] : aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K)) ).

% numeral_eq_Suc
tff(fact_7743_abs__div,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y),Xb)
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Y)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,Xb)),aa_int_int(abs_abs_int,Y)) ) ) ).

% abs_div
tff(fact_7744_suminf__add,axiom,
    ! [Fa: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(Fa)
     => ( summable_complex(G)
       => ( aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,suminf_complex(Fa)),suminf_complex(G)) = suminf_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vb(Fa),G)) ) ) ) ).

% suminf_add
tff(fact_7745_suminf__add,axiom,
    ! [Fa: fun_nat_real,G: fun_nat_real] :
      ( summable_real(Fa)
     => ( summable_real(G)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,suminf_real(Fa)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_vc(Fa),G)) ) ) ) ).

% suminf_add
tff(fact_7746_suminf__add,axiom,
    ! [Fa: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(Fa)
     => ( summable_nat(G)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,suminf_nat(Fa)),suminf_nat(G)) = suminf_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_vd(Fa),G)) ) ) ) ).

% suminf_add
tff(fact_7747_suminf__add,axiom,
    ! [Fa: fun_nat_int,G: fun_nat_int] :
      ( summable_int(Fa)
     => ( summable_int(G)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,suminf_int(Fa)),suminf_int(G)) = suminf_int(aa_fun5565043122120104496at_int(aTP_Lamm_ve(Fa),G)) ) ) ) ).

% suminf_add
tff(fact_7748_suminf__mult,axiom,
    ! [Fa: fun_nat_complex,C: complex] :
      ( summable_complex(Fa)
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_vh(Fa),C)) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,C),suminf_complex(Fa)) ) ) ).

% suminf_mult
tff(fact_7749_suminf__mult,axiom,
    ! [Fa: fun_nat_real,C: real] :
      ( summable_real(Fa)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_vi(Fa),C)) = aa_real_real(times_times_real(C),suminf_real(Fa)) ) ) ).

% suminf_mult
tff(fact_7750_suminf__mult2,axiom,
    ! [Fa: fun_nat_complex,C: complex] :
      ( summable_complex(Fa)
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,suminf_complex(Fa)),C) = suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_vf(Fa),C)) ) ) ).

% suminf_mult2
tff(fact_7751_suminf__mult2,axiom,
    ! [Fa: fun_nat_real,C: real] :
      ( summable_real(Fa)
     => ( aa_real_real(times_times_real(suminf_real(Fa)),C) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_vg(Fa),C)) ) ) ).

% suminf_mult2
tff(fact_7752_suminf__diff,axiom,
    ! [Fa: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(Fa)
     => ( summable_complex(G)
       => ( aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,suminf_complex(Fa)),suminf_complex(G)) = suminf_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vj(Fa),G)) ) ) ) ).

% suminf_diff
tff(fact_7753_suminf__diff,axiom,
    ! [Fa: fun_nat_real,G: fun_nat_real] :
      ( summable_real(Fa)
     => ( summable_real(G)
       => ( aa_real_real(minus_minus_real(suminf_real(Fa)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_vk(Fa),G)) ) ) ) ).

% suminf_diff
tff(fact_7754_suminf__divide,axiom,
    ! [Fa: fun_nat_complex,C: complex] :
      ( summable_complex(Fa)
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_uv(Fa),C)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,suminf_complex(Fa)),C) ) ) ).

% suminf_divide
tff(fact_7755_suminf__divide,axiom,
    ! [Fa: fun_nat_real,C: real] :
      ( summable_real(Fa)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_uw(Fa),C)) = aa_real_real(divide_divide_real(suminf_real(Fa)),C) ) ) ).

% suminf_divide
tff(fact_7756_suminf__minus,axiom,
    ! [Fa: fun_nat_complex] :
      ( summable_complex(Fa)
     => ( suminf_complex(aTP_Lamm_vn(Fa)) = aa_complex_complex(uminus1482373934393186551omplex,suminf_complex(Fa)) ) ) ).

% suminf_minus
tff(fact_7757_suminf__minus,axiom,
    ! [Fa: fun_nat_real] :
      ( summable_real(Fa)
     => ( suminf_real(aTP_Lamm_vo(Fa)) = aa_real_real(uminus_uminus_real,suminf_real(Fa)) ) ) ).

% suminf_minus
tff(fact_7758_cos__arctan__not__zero,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(arctan,Xb)) != zero_zero_real ).

% cos_arctan_not_zero
tff(fact_7759_suminf__nonneg,axiom,
    ! [Fa: fun_nat_real] :
      ( summable_real(Fa)
     => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Fa,N3))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),suminf_real(Fa)) ) ) ).

% suminf_nonneg
tff(fact_7760_suminf__nonneg,axiom,
    ! [Fa: fun_nat_nat] :
      ( summable_nat(Fa)
     => ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(Fa,N3))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),suminf_nat(Fa)) ) ) ).

% suminf_nonneg
tff(fact_7761_suminf__nonneg,axiom,
    ! [Fa: fun_nat_int] :
      ( summable_int(Fa)
     => ( ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(Fa,N3))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),suminf_int(Fa)) ) ) ).

% suminf_nonneg
tff(fact_7762_suminf__eq__zero__iff,axiom,
    ! [Fa: fun_nat_real] :
      ( summable_real(Fa)
     => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Fa,N3))
       => ( ( suminf_real(Fa) = zero_zero_real )
        <=> ! [N8: nat] : aa_nat_real(Fa,N8) = zero_zero_real ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7763_suminf__eq__zero__iff,axiom,
    ! [Fa: fun_nat_nat] :
      ( summable_nat(Fa)
     => ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(Fa,N3))
       => ( ( suminf_nat(Fa) = zero_zero_nat )
        <=> ! [N8: nat] : aa_nat_nat(Fa,N8) = zero_zero_nat ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7764_suminf__eq__zero__iff,axiom,
    ! [Fa: fun_nat_int] :
      ( summable_int(Fa)
     => ( ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(Fa,N3))
       => ( ( suminf_int(Fa) = zero_zero_int )
        <=> ! [N8: nat] : aa_nat_int(Fa,N8) = zero_zero_int ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7765_suminf__pos,axiom,
    ! [Fa: fun_nat_real] :
      ( summable_real(Fa)
     => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(Fa,N3))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(Fa)) ) ) ).

% suminf_pos
tff(fact_7766_suminf__pos,axiom,
    ! [Fa: fun_nat_nat] :
      ( summable_nat(Fa)
     => ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(Fa,N3))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(Fa)) ) ) ).

% suminf_pos
tff(fact_7767_suminf__pos,axiom,
    ! [Fa: fun_nat_int] :
      ( summable_int(Fa)
     => ( ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(Fa,N3))
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(Fa)) ) ) ).

% suminf_pos
tff(fact_7768_summable__zero__power_H,axiom,
    ! [Fa: fun_nat_complex] : summable_complex(aTP_Lamm_tu(Fa)) ).

% summable_zero_power'
tff(fact_7769_summable__zero__power_H,axiom,
    ! [Fa: fun_nat_real] : summable_real(aTP_Lamm_tv(Fa)) ).

% summable_zero_power'
tff(fact_7770_summable__zero__power_H,axiom,
    ! [Fa: fun_nat_int] : summable_int(aTP_Lamm_vu(Fa)) ).

% summable_zero_power'
tff(fact_7771_summable__0__powser,axiom,
    ! [Fa: fun_nat_complex] : summable_complex(aTP_Lamm_tu(Fa)) ).

% summable_0_powser
tff(fact_7772_summable__0__powser,axiom,
    ! [Fa: fun_nat_real] : summable_real(aTP_Lamm_tv(Fa)) ).

% summable_0_powser
tff(fact_7773_powser__split__head_I3_J,axiom,
    ! [Fa: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_vs(Fa),Z))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_vv(Fa),Z)) ) ).

% powser_split_head(3)
tff(fact_7774_powser__split__head_I3_J,axiom,
    ! [Fa: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_vq(Fa),Z))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_vw(Fa),Z)) ) ).

% powser_split_head(3)
tff(fact_7775_summable__powser__split__head,axiom,
    ! [Fa: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_vv(Fa),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_vs(Fa),Z)) ) ).

% summable_powser_split_head
tff(fact_7776_summable__powser__split__head,axiom,
    ! [Fa: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_vw(Fa),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_vq(Fa),Z)) ) ).

% summable_powser_split_head
tff(fact_7777_summable__powser__ignore__initial__segment,axiom,
    ! [Fa: fun_nat_complex,M: nat,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_vx(Fa),M),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_vs(Fa),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7778_summable__powser__ignore__initial__segment,axiom,
    ! [Fa: fun_nat_real,M: nat,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_vy(Fa),M),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_vq(Fa),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7779_zabs__def,axiom,
    ! [I: int] :
      aa_int_int(abs_abs_int,I) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int),aa_int_int(uminus_uminus_int,I),I) ).

% zabs_def
tff(fact_7780_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))),aa_int_int(abs_abs_int,L)) ) ).

% abs_mod_less
tff(fact_7781_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),I)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I)) ) ) ).

% dvd_imp_le_int
tff(fact_7782_summable__norm__comparison__test,axiom,
    ! [Fa: fun_nat_complex,G: fun_nat_real] :
      ( ? [N9: nat] :
        ! [N3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N9),N3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Fa,N3))),aa_nat_real(G,N3)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_uy(Fa)) ) ) ).

% summable_norm_comparison_test
tff(fact_7783_summable__rabs__comparison__test,axiom,
    ! [Fa: fun_nat_real,G: fun_nat_real] :
      ( ? [N9: nat] :
        ! [N3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N9),N3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(Fa,N3))),aa_nat_real(G,N3)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_vp(Fa)) ) ) ).

% summable_rabs_comparison_test
tff(fact_7784_pred__numeral__def,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),one_one_nat) ).

% pred_numeral_def
tff(fact_7785_abs__integer__code,axiom,
    ! [K: code_integer] :
      aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),K) ).

% abs_integer_code
tff(fact_7786_summable__rabs,axiom,
    ! [Fa: fun_nat_real] :
      ( summable_real(aTP_Lamm_vp(Fa))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(Fa))),suminf_real(aTP_Lamm_vp(Fa))) ) ).

% summable_rabs
tff(fact_7787_suminf__pos2,axiom,
    ! [Fa: fun_nat_real,I: nat] :
      ( summable_real(Fa)
     => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Fa,N3))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(Fa,I))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(Fa)) ) ) ) ).

% suminf_pos2
tff(fact_7788_suminf__pos2,axiom,
    ! [Fa: fun_nat_nat,I: nat] :
      ( summable_nat(Fa)
     => ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(Fa,N3))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(Fa,I))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(Fa)) ) ) ) ).

% suminf_pos2
tff(fact_7789_suminf__pos2,axiom,
    ! [Fa: fun_nat_int,I: nat] :
      ( summable_int(Fa)
     => ( ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(Fa,N3))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(Fa,I))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(Fa)) ) ) ) ).

% suminf_pos2
tff(fact_7790_suminf__pos__iff,axiom,
    ! [Fa: fun_nat_real] :
      ( summable_real(Fa)
     => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Fa,N3))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(Fa))
        <=> ? [I4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(Fa,I4)) ) ) ) ).

% suminf_pos_iff
tff(fact_7791_suminf__pos__iff,axiom,
    ! [Fa: fun_nat_nat] :
      ( summable_nat(Fa)
     => ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(Fa,N3))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(Fa))
        <=> ? [I4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(Fa,I4)) ) ) ) ).

% suminf_pos_iff
tff(fact_7792_suminf__pos__iff,axiom,
    ! [Fa: fun_nat_int] :
      ( summable_int(Fa)
     => ( ! [N3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(Fa,N3))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(Fa))
        <=> ? [I4: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(Fa,I4)) ) ) ) ).

% suminf_pos_iff
tff(fact_7793_powser__inside,axiom,
    ! [Fa: fun_nat_real,Xb: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_vq(Fa),Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,Xb))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_vq(Fa),Z)) ) ) ).

% powser_inside
tff(fact_7794_powser__inside,axiom,
    ! [Fa: fun_nat_complex,Xb: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_vs(Fa),Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,Xb))
       => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_vs(Fa),Z)) ) ) ).

% powser_inside
tff(fact_7795_suminf__split__head,axiom,
    ! [Fa: fun_nat_complex] :
      ( summable_complex(Fa)
     => ( suminf_complex(aTP_Lamm_vm(Fa)) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,suminf_complex(Fa)),aa_nat_complex(Fa,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_7796_suminf__split__head,axiom,
    ! [Fa: fun_nat_real] :
      ( summable_real(Fa)
     => ( suminf_real(aTP_Lamm_vl(Fa)) = aa_real_real(minus_minus_real(suminf_real(Fa)),aa_nat_real(Fa,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_7797_complete__algebra__summable__geometric,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),one_one_real)
     => summable_real(aa_real_fun_nat_real(power_power_real,Xb)) ) ).

% complete_algebra_summable_geometric
tff(fact_7798_complete__algebra__summable__geometric,axiom,
    ! [Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),one_one_real)
     => summable_complex(aa_com8181001469867462503omplex(power_power_complex,Xb)) ) ).

% complete_algebra_summable_geometric
tff(fact_7799_summable__geometric,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => summable_real(aa_real_fun_nat_real(power_power_real,C)) ) ).

% summable_geometric
tff(fact_7800_summable__geometric,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => summable_complex(aa_com8181001469867462503omplex(power_power_complex,C)) ) ).

% summable_geometric
tff(fact_7801_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,M),N)),M)
      <=> ( aa_int_int(abs_abs_int,N) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_7802_summable__norm,axiom,
    ! [Fa: fun_nat_real] :
      ( summable_real(aTP_Lamm_ux(Fa))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(Fa))),suminf_real(aTP_Lamm_ux(Fa))) ) ).

% summable_norm
tff(fact_7803_summable__norm,axiom,
    ! [Fa: fun_nat_complex] :
      ( summable_real(aTP_Lamm_uy(Fa))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(Fa))),suminf_real(aTP_Lamm_uy(Fa))) ) ).

% summable_norm
tff(fact_7804_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,K)),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ) ).

% even_abs_add_iff
tff(fact_7805_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(abs_abs_int,L)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ) ).

% even_add_abs_iff
tff(fact_7806_powser__split__head_I1_J,axiom,
    ! [Fa: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_vs(Fa),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_vs(Fa),Z)) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(Fa,zero_zero_nat)),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_vv(Fa),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_7807_powser__split__head_I1_J,axiom,
    ! [Fa: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_vq(Fa),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_vq(Fa),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Fa,zero_zero_nat)),aa_real_real(times_times_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_vw(Fa),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_7808_powser__split__head_I2_J,axiom,
    ! [Fa: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_vs(Fa),Z))
     => ( aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_vv(Fa),Z))),Z) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_vs(Fa),Z))),aa_nat_complex(Fa,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7809_powser__split__head_I2_J,axiom,
    ! [Fa: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_vq(Fa),Z))
     => ( aa_real_real(times_times_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_vw(Fa),Z))),Z) = aa_real_real(minus_minus_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_vq(Fa),Z))),aa_nat_real(Fa,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7810_monoseq__realpow,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,Xb)) ) ) ).

% monoseq_realpow
tff(fact_7811_suminf__exist__split,axiom,
    ! [R: real,Fa: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
     => ( summable_real(Fa)
       => ? [N10: nat] :
          ! [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N10),N4)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_ur(Fa),N4)))),R) ) ) ) ).

% suminf_exist_split
tff(fact_7812_suminf__exist__split,axiom,
    ! [R: real,Fa: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
     => ( summable_complex(Fa)
       => ? [N10: nat] :
          ! [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N10),N4)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_us(Fa),N4)))),R) ) ) ) ).

% suminf_exist_split
tff(fact_7813_summable__power__series,axiom,
    ! [Fa: fun_nat_real,Z: real] :
      ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Fa,I2)),one_one_real)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Fa,I2))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Z)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),one_one_real)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_vq(Fa),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_7814_Abel__lemma,axiom,
    ! [R: real,R0: real,A2: fun_nat_complex,M8: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,R),R0)
       => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(A2,N3))),aa_nat_real(aa_real_fun_nat_real(power_power_real,R0),N3))),M8)
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_vz(R),A2)) ) ) ) ).

% Abel_lemma
tff(fact_7815_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,Fa: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),N) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Fa,aa_nat_nat(suc,I2))),aa_nat_int(Fa,I2)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,M)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(Fa,N))
           => ? [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N)
                & ( aa_nat_int(Fa,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_7816_decr__lemma,axiom,
    ! [D: int,Xb: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Z))),one_one_int)),D))),Z) ) ).

% decr_lemma
tff(fact_7817_incr__lemma,axiom,
    ! [D: int,Z: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Z))),one_one_int)),D))) ) ).

% incr_lemma
tff(fact_7818_summable__ratio__test,axiom,
    ! [C: real,N2: nat,Fa: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),one_one_real)
     => ( ! [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Fa,aa_nat_nat(suc,N3)))),aa_real_real(times_times_real(C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Fa,N3)))) )
       => summable_real(Fa) ) ) ).

% summable_ratio_test
tff(fact_7819_summable__ratio__test,axiom,
    ! [C: real,N2: nat,Fa: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),one_one_real)
     => ( ! [N3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Fa,aa_nat_nat(suc,N3)))),aa_real_real(times_times_real(C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Fa,N3)))) )
       => summable_complex(Fa) ) ) ).

% summable_ratio_test
tff(fact_7820_arctan__ubound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% arctan_ubound
tff(fact_7821_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% arctan_one
tff(fact_7822_nat__ivt__aux,axiom,
    ! [N: nat,Fa: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Fa,aa_nat_nat(suc,I2))),aa_nat_int(Fa,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(Fa,N))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N)
              & ( aa_nat_int(Fa,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_7823_arctan__lbound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y)) ).

% arctan_lbound
tff(fact_7824_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% arctan_bounded
tff(fact_7825_nat0__intermed__int__val,axiom,
    ! [N: nat,Fa: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Fa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(Fa,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(Fa,N))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N)
              & ( aa_nat_int(Fa,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_7826_arctan__add,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(Xb),Y)))) ) ) ) ).

% arctan_add
tff(fact_7827_machin__Euler,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(divide_divide_real(numeral_numeral_real(aa_num_num(bit1,one))),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_7828_machin,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))))))) ).

% machin
tff(fact_7829_arctan__double,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(arctan,Xb)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),Xb)),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_7830_tan__double,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(tan_complex,Xb))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(tan_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_7831_tan__double,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),Xb)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),Xb)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(tan_real,Xb))),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_7832_even__word__def,axiom,
    even_w9054469088133485505l_num1 = dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))) ).

% even_word_def
tff(fact_7833_triangle__def,axiom,
    ! [N: nat] : aa_nat_nat(nat_triangle,N) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_7834_exE__realizer,axiom,
    ! [P: fun_nat_fun_nat_o,P2: product_prod_nat_nat,Q: fun_Pr4796943543331071514_nat_o,Fa: fun_na4138050610385705875at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(P,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2)),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2))
     => ( ! [X2: nat,Y2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P,Y2),X2)
           => aa_Pro6150637604445964947_nat_o(Q,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(Fa,X2),Y2)) )
       => aa_Pro6150637604445964947_nat_o(Q,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(Fa),P2)) ) ) ).

% exE_realizer
tff(fact_7835_exE__realizer,axiom,
    ! [P: fun_nat_fun_nat_o,P2: product_prod_nat_nat,Q: fun_o_o,Fa: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(P,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2)),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2))
     => ( ! [X2: nat,Y2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P,Y2),X2)
           => aa_o_o(Q,aa_nat_o(aa_nat_fun_nat_o(Fa,X2),Y2)) )
       => aa_o_o(Q,aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(Fa),P2)) ) ) ).

% exE_realizer
tff(fact_7836_exE__realizer,axiom,
    ! [P: fun_int_fun_int_o,P2: product_prod_int_int,Q: fun_Pr3238374608701176162_int_o,Fa: fun_in1696471567231710611nt_int] :
      ( aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P2))
     => ( ! [X2: int,Y2: int] :
            ( aa_int_o(aa_int_fun_int_o(P,Y2),X2)
           => aa_Pro4592068669816069595_int_o(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Fa,X2),Y2)) )
       => aa_Pro4592068669816069595_int_o(Q,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(Fa),P2)) ) ) ).

% exE_realizer
tff(fact_7837_exE__realizer,axiom,
    ! [P: fun_int_fun_int_o,P2: product_prod_int_int,Q: fun_o_o,Fa: fun_int_fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P2))
     => ( ! [X2: int,Y2: int] :
            ( aa_int_o(aa_int_fun_int_o(P,Y2),X2)
           => aa_o_o(Q,aa_int_o(aa_int_fun_int_o(Fa,X2),Y2)) )
       => aa_o_o(Q,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Fa),P2)) ) ) ).

% exE_realizer
tff(fact_7838_exE__realizer,axiom,
    ! [P: fun_int_fun_int_o,P2: product_prod_int_int,Q: fun_int_o,Fa: fun_int_fun_int_int] :
      ( aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P2))
     => ( ! [X2: int,Y2: int] :
            ( aa_int_o(aa_int_fun_int_o(P,Y2),X2)
           => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(Fa,X2),Y2)) )
       => aa_int_o(Q,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(Fa),P2)) ) ) ).

% exE_realizer
tff(fact_7839_VEBT__internal_Ovebt__inserti_H_Ofixp__induct,axiom,
    ! [P: fun_fu1421233379579251956EBTi_o] :
      ( comple8556566605269262190_VEBTi(partia7782936097874681665Ti_nat(heap_T3112222404744780921_VEBTi),partia1868168049876374393Ti_nat(heap_T7173139186834293313_VEBTi),aTP_Lamm_wa(P))
     => ( aa_fun2485637723310249309EBTi_o(P,aTP_Lamm_wb)
       => ( ! [F3: fun_VE2018681615261037117_VEBTi] :
              ( aa_fun2485637723310249309EBTi_o(P,F3)
             => aa_fun2485637723310249309EBTi_o(P,aTP_Lamm_xc(F3)) )
         => aa_fun2485637723310249309EBTi_o(P,vEBT_V3964819847710782039nserti) ) ) ) ).

% VEBT_internal.vebt_inserti'.fixp_induct
tff(fact_7840_TBOUND__VEBT__case,axiom,
    ! [Tib: vEBT_VEBTi,Fa: fun_o_7009304396571837746ap_nat,Bnd: fun_o_fun_o_nat,F: fun_op2425996437410981688ap_nat,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A: $o,B: $o] :
          ( ( Tib = vEBT_Leafi((A),(B)) )
         => aa_nat_o(time_TBOUND_nat(aa_o_H8739796202006397184ap_nat(aa_o_f2078835126877633507ap_nat(Fa,(A)),(B))),aa_o_nat(aa_o_fun_o_nat(Bnd,(A)),(B))) )
     => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] :
            ( ( Tib = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info,Deg,TreeArray),Summary) )
           => aa_nat_o(time_TBOUND_nat(aa_VEB183921879236458153ap_nat(aa_arr6934346397072338570ap_nat(aa_nat1288005622339309855ap_nat(aa_opt6697879614838309225ap_nat(F,Info),Deg),TreeArray),Summary)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info),Deg),TreeArray),Summary)) )
       => aa_nat_o(time_TBOUND_nat(vEBT_c1335663792808957512ap_nat(F,Fa,Tib)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Tib)) ) ) ).

% TBOUND_VEBT_case
tff(fact_7841_TBOUND__VEBT__case,axiom,
    ! [Tib: vEBT_VEBTi,Fa: fun_o_4049874799390722306on_nat,Bnd: fun_o_fun_o_nat,F: fun_op5783775701855621896on_nat,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A: $o,B: $o] :
          ( ( Tib = vEBT_Leafi((A),(B)) )
         => aa_nat_o(time_T8353473612707095248on_nat(aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(Fa,(A)),(B))),aa_o_nat(aa_o_fun_o_nat(Bnd,(A)),(B))) )
     => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] :
            ( ( Tib = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info,Deg,TreeArray),Summary) )
           => aa_nat_o(time_T8353473612707095248on_nat(aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F,Info),Deg),TreeArray),Summary)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info),Deg),TreeArray),Summary)) )
       => aa_nat_o(time_T8353473612707095248on_nat(aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(F,Fa),Tib)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Tib)) ) ) ).

% TBOUND_VEBT_case
tff(fact_7842_TBOUND__VEBT__case,axiom,
    ! [Tib: vEBT_VEBTi,Fa: fun_o_1207901889202581752Heap_o,Bnd: fun_o_fun_o_nat,F: fun_op1763656958002597022Heap_o,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A: $o,B: $o] :
          ( ( Tib = vEBT_Leafi((A),(B)) )
         => aa_nat_o(time_TBOUND_o(aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(Fa,(A)),(B))),aa_o_nat(aa_o_fun_o_nat(Bnd,(A)),(B))) )
     => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] :
            ( ( Tib = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info,Deg,TreeArray),Summary) )
           => aa_nat_o(time_TBOUND_o(aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(F,Info),Deg),TreeArray),Summary)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info),Deg),TreeArray),Summary)) )
       => aa_nat_o(time_TBOUND_o(aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(F,Fa),Tib)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Tib)) ) ) ).

% TBOUND_VEBT_case
tff(fact_7843_TBOUND__VEBT__case,axiom,
    ! [Tib: vEBT_VEBTi,Fa: fun_o_6972675622308511895_VEBTi,Bnd: fun_o_fun_o_nat,F: fun_op5127434927081926045_VEBTi,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A: $o,B: $o] :
          ( ( Tib = vEBT_Leafi((A),(B)) )
         => aa_nat_o(time_T5737551269749752165_VEBTi(aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(Fa,(A)),(B))),aa_o_nat(aa_o_fun_o_nat(Bnd,(A)),(B))) )
     => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] :
            ( ( Tib = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info,Deg,TreeArray),Summary) )
           => aa_nat_o(time_T5737551269749752165_VEBTi(aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F,Info),Deg),TreeArray),Summary)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info),Deg),TreeArray),Summary)) )
       => aa_nat_o(time_T5737551269749752165_VEBTi(aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(F,Fa),Tib)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Tib)) ) ) ).

% TBOUND_VEBT_case
tff(fact_7844_tan__pi,axiom,
    aa_real_real(tan_real,pi) = zero_zero_real ).

% tan_pi
tff(fact_7845_tan__zero,axiom,
    aa_real_real(tan_real,zero_zero_real) = zero_zero_real ).

% tan_zero
tff(fact_7846_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_7847_tan__npi,axiom,
    ! [N: nat] : aa_real_real(tan_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_7848_tan__periodic,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic
tff(fact_7849_VEBTi_Osimps_I5_J,axiom,
    ! [F1: fun_op5783775701855621896on_nat,F2: fun_o_4049874799390722306on_nat,X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(F1,F2),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X12,X13),X14)) = aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F1,X11),X12),X13),X14) ).

% VEBTi.simps(5)
tff(fact_7850_VEBTi_Osimps_I5_J,axiom,
    ! [F1: fun_op1763656958002597022Heap_o,F2: fun_o_1207901889202581752Heap_o,X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(F1,F2),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X12,X13),X14)) = aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(F1,X11),X12),X13),X14) ).

% VEBTi.simps(5)
tff(fact_7851_VEBTi_Osimps_I5_J,axiom,
    ! [F1: fun_op5127434927081926045_VEBTi,F2: fun_o_6972675622308511895_VEBTi,X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(F1,F2),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X12,X13),X14)) = aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F1,X11),X12),X13),X14) ).

% VEBTi.simps(5)
tff(fact_7852_VEBTi_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,vEBT_Leafi((X21),(X222))) = zero_zero_nat ).

% VEBTi.size(4)
tff(fact_7853_vebt__assn__raw_Ocases,axiom,
    ! [Xb: produc3625547720036274456_VEBTi] :
      ( ! [A: $o,B: $o,Ai: $o,Bi: $o] : Xb != produc6084888613844515218_VEBTi(vEBT_Leaf((A),(B)),vEBT_Leafi((Ai),(Bi)))
     => ( ! [Mmo: option4927543243414619207at_nat,Deg: nat,Tree_list: list_VEBT_VEBT,Summary: vEBT_VEBT,Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] : Xb != produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg,Tree_list),Summary),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi))
       => ( ! [V2: option4927543243414619207at_nat,Va2: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT,Vd3: $o,Ve3: $o] : Xb != produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va2,Vb3),Vc3),vEBT_Leafi((Vd3),(Ve3)))
         => ~ ! [Vd3: $o,Ve3: $o,V2: option4927543243414619207at_nat,Va2: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] : Xb != produc6084888613844515218_VEBTi(vEBT_Leaf((Vd3),(Ve3)),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va2,Vb3),Vc3)) ) ) ) ).

% vebt_assn_raw.cases
tff(fact_7854_VEBT__internal_OminNulli_Ocases,axiom,
    ! [Xb: vEBT_VEBTi] :
      ( ( Xb != vEBT_Leafi($false,$false) )
     => ( ! [Uv: $o] : Xb != vEBT_Leafi($true,(Uv))
       => ( ! [Uu2: $o] : Xb != vEBT_Leafi((Uu2),$true)
         => ( ! [Uw: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] : Xb != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw,Ux2),Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] : Xb != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2) ) ) ) ) ).

% VEBT_internal.minNulli.cases
tff(fact_7855_VEBTi_Osimps_I6_J,axiom,
    ! [F1: fun_op5783775701855621896on_nat,F2: fun_o_4049874799390722306on_nat,X21: $o,X222: $o] : aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(F1,F2),vEBT_Leafi((X21),(X222))) = aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(F2,(X21)),(X222)) ).

% VEBTi.simps(6)
tff(fact_7856_VEBTi_Osimps_I6_J,axiom,
    ! [F1: fun_op1763656958002597022Heap_o,F2: fun_o_1207901889202581752Heap_o,X21: $o,X222: $o] : aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(F1,F2),vEBT_Leafi((X21),(X222))) = aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(F2,(X21)),(X222)) ).

% VEBTi.simps(6)
tff(fact_7857_VEBTi_Osimps_I6_J,axiom,
    ! [F1: fun_op5127434927081926045_VEBTi,F2: fun_o_6972675622308511895_VEBTi,X21: $o,X222: $o] : aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(F1,F2),vEBT_Leafi((X21),(X222))) = aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(F2,(X21)),(X222)) ).

% VEBTi.simps(6)
tff(fact_7858_vebt__minti_Ocases,axiom,
    ! [Xb: vEBT_VEBTi] :
      ( ! [A: $o,B: $o] : Xb != vEBT_Leafi((A),(B))
     => ( ! [Uu2: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] : Xb != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv),Uw)
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] : Xb != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2) ) ) ).

% vebt_minti.cases
tff(fact_7859_TBOUND__upd,axiom,
    ! [Xs: nat,I: vEBT_VEBTi,Xb: array_VEBT_VEBTi] : aa_nat_o(time_T6070283812100419266_VEBTi(array_upd_VEBT_VEBTi(Xs,I,Xb)),one_one_nat) ).

% TBOUND_upd
tff(fact_7860_VEBT__internal_OminNulli_Oelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_Time_Heap_o] :
      ( ( aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Xb) = Y )
     => ( ( ( Xb = vEBT_Leafi($false,$false) )
         => ( Y != aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true) ) )
       => ( ( ? [Uv: $o] : Xb = vEBT_Leafi($true,(Uv))
           => ( Y != aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) ) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leafi((Uu2),$true)
             => ( Y != aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) ) )
           => ( ( ? [Uw: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] : Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw,Ux2),Uy2)
               => ( Y != aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true) ) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] : Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2)
                 => ( Y != aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.elims
tff(fact_7861_VEBT__internal_Ovebt__buildupi_H_Osimps_I1_J,axiom,
    aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,zero_zero_nat) = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) ).

% VEBT_internal.vebt_buildupi'.simps(1)
tff(fact_7862_vebt__buildupi_Osimps_I1_J,axiom,
    aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,zero_zero_nat) = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) ).

% vebt_buildupi.simps(1)
tff(fact_7863_VEBT__internal_Ovebt__buildupi_H_Osimps_I2_J,axiom,
    aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,aa_nat_nat(suc,zero_zero_nat)) = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) ).

% VEBT_internal.vebt_buildupi'.simps(2)
tff(fact_7864_vebt__buildupi_Osimps_I2_J,axiom,
    aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,aa_nat_nat(suc,zero_zero_nat)) = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) ).

% vebt_buildupi.simps(2)
tff(fact_7865_VEBT__internal_OminNulli_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: array_VEBT_VEBTi,Vc: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc)) = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) ).

% VEBT_internal.minNulli.simps(5)
tff(fact_7866_tan__def,axiom,
    ! [X3: real] : aa_real_real(tan_real,X3) = aa_real_real(divide_divide_real(aa_real_real(sin_real,X3)),aa_real_real(cos_real,X3)) ).

% tan_def
tff(fact_7867_vebt__maxti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv2),Uw2)) = heap_T3487192422709364219on_nat(none_nat) ).

% vebt_maxti.simps(2)
tff(fact_7868_vebt__minti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv2),Uw2)) = heap_T3487192422709364219on_nat(none_nat) ).

% vebt_minti.simps(2)
tff(fact_7869_vebt__maxti_Oelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Xb) = Y )
     => ( ! [A: $o,B: $o] :
            ( ( Xb = vEBT_Leafi((A),(B)) )
           => ( Y != $ite(
                  (B),
                  heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),
                  $ite((A),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] : Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv),Uw)
           => ( Y != heap_T3487192422709364219on_nat(none_nat) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] : Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2)
               => ( Y != heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,Ma2)) ) ) ) ) ) ).

% vebt_maxti.elims
tff(fact_7870_vebt__minti_Oelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Xb) = Y )
     => ( ! [A: $o,B: $o] :
            ( ( Xb = vEBT_Leafi((A),(B)) )
           => ( Y != $ite(
                  (A),
                  heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),
                  $ite((B),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),heap_T3487192422709364219on_nat(none_nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] : Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv),Uw)
           => ( Y != heap_T3487192422709364219on_nat(none_nat) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] : Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2)
               => ( Y != heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,Mi2)) ) ) ) ) ) ).

% vebt_minti.elims
tff(fact_7871_vebt__maxti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux,Uy),Uz)) = heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,Ma)) ).

% vebt_maxti.simps(3)
tff(fact_7872_vebt__minti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Ux,Uy),Uz)) = heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,Mi)) ).

% vebt_minti.simps(3)
tff(fact_7873_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = one_one_real ).

% tan_45
tff(fact_7874_vebt__maxti_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,vEBT_Leafi((A2),(B2))) = $ite(
        (B2),
        heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),
        $ite((A2),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)) ) ).

% vebt_maxti.simps(1)
tff(fact_7875_vebt__minti_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,vEBT_Leafi((A2),(B2))) = $ite(
        (A2),
        heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),
        $ite((B2),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),heap_T3487192422709364219on_nat(none_nat)) ) ).

% vebt_minti.simps(1)
tff(fact_7876_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(tan_real,X2)) ) ) ).

% lemma_tan_total
tff(fact_7877_tan__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tan_real,Xb)) ) ) ).

% tan_gt_zero
tff(fact_7878_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X2) = Y ) ) ).

% lemma_tan_total1
tff(fact_7879_tan__mono__lt__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_7880_tan__monotone_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xb)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_7881_tan__monotone,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xb)) ) ) ) ).

% tan_monotone
tff(fact_7882_tan__total,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),X2)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X2) = Y )
      & ! [Y4: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
            & ( aa_real_real(tan_real,Y4) = Y ) )
         => ( Y4 = X2 ) ) ) ).

% tan_total
tff(fact_7883_tan__minus__45,axiom,
    aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_7884_tan__inverse,axiom,
    ! [Y: real] : aa_real_real(divide_divide_real(one_one_real),aa_real_real(tan_real,Y)) = aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))),Y)) ).

% tan_inverse
tff(fact_7885_vebt__inserti_Omono,axiom,
    ! [Xb: produc3881548065746020326Ti_nat] : comple6664421360520557306_VEBTi(partia6690842624828592406Ti_nat(heap_T7173139186834293313_VEBTi),heap_T7173139186834293313_VEBTi,aTP_Lamm_xr(Xb)) ).

% vebt_inserti.mono
tff(fact_7886_add__tan__eq,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)) = aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))) ) ) ) ).

% add_tan_eq
tff(fact_7887_exE__realizer_H,axiom,
    ! [P: fun_nat_fun_nat_o,P2: product_prod_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(P,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2)),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2))
     => ~ ! [X2: nat,Y2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(P,Y2),X2) ) ).

% exE_realizer'
tff(fact_7888_exE__realizer_H,axiom,
    ! [P: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),aa_Pro6799258434465214121nt_int(product_fst_int_int,P2))
     => ~ ! [X2: int,Y2: int] : ~ aa_int_o(aa_int_fun_int_o(P,Y2),X2) ) ).

% exE_realizer'
tff(fact_7889_exE__realizer_H,axiom,
    ! [P: fun_assn_fun_assn_o,P2: produc6575502325842934193n_assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(P,aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,P2)),aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,P2))
     => ~ ! [X2: assn,Y2: assn] : ~ aa_assn_o(aa_assn_fun_assn_o(P,Y2),X2) ) ).

% exE_realizer'
tff(fact_7890_vebt__inserti_Oraw__induct,axiom,
    ! [P: fun_VE8508601879845319688_nat_o,Xaa: produc3881548065746020326Ti_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: vEBT_VEBTi,N: nat] :
      ( ! [Vebt_inserti: fun_VE8118064890490510894_VEBTi] :
          ( ! [A5: vEBT_VEBTi,B4: nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R2: vEBT_VEBTi,N4: nat] :
              ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(Vebt_inserti,A5),B4),H3,H4),R2),N4)
             => aa_nat_o(aa_VEB8883426185042993134_nat_o(aa_hea5256493101677210548_nat_o(aa_hea3862133550498874060_nat_o(aa_nat3537556108769025421_nat_o(aa_VEB2168324858650567793_nat_o(P,A5),B4),H3),H4),R2),N4) )
         => ! [T: vEBT_VEBTi,X2: nat,Ta: heap_e7401611519738050253t_unit,Xa2: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N3: nat] :
              ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_nat3009484633486537743_VEBTi(aTP_Lamm_ye(Vebt_inserti),X2),aTP_Lamm_xb(X2)),T),Ta,Xa2),R3),N3)
             => aa_nat_o(aa_VEB8883426185042993134_nat_o(aa_hea5256493101677210548_nat_o(aa_hea3862133550498874060_nat_o(aa_nat3537556108769025421_nat_o(aa_VEB2168324858650567793_nat_o(P,T),X2),Ta),Xa2),R3),N3) ) )
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_Pro1482084341531967383_VEBTi(produc3255295512018472142_VEBTi(vEBT_vebt_inserti),Xaa),H,H2),R),N)
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(aa_hea5256493101677210548_nat_o(aa_hea3862133550498874060_nat_o(produc2033849612716375976_nat_o(P,Xaa),H),H2),R),N) ) ) ).

% vebt_inserti.raw_induct
tff(fact_7891_vebt__inserti_Osimps,axiom,
    ! [Tb: vEBT_VEBTi,Xb: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(vEBT_vebt_inserti,Tb),Xb) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aTP_Lamm_yr(Xb),aTP_Lamm_xb(Xb)),Tb) ).

% vebt_inserti.simps
tff(fact_7892_vebt__inserti_Ofixp__induct,axiom,
    ! [P: fun_fu1178290271985267629EBTi_o] :
      ( comple7878442886591315089_VEBTi(partia6972460264168101086Ti_nat(heap_T3112222404744780921_VEBTi),partia6690842624828592406Ti_nat(heap_T7173139186834293313_VEBTi),aTP_Lamm_ys(P))
     => ( aa_fun6333388201673717798EBTi_o(P,aTP_Lamm_yt)
       => ( ! [F3: fun_VE8118064890490510894_VEBTi] :
              ( aa_fun6333388201673717798EBTi_o(P,F3)
             => aa_fun6333388201673717798EBTi_o(P,aTP_Lamm_yu(F3)) )
         => aa_fun6333388201673717798EBTi_o(P,vEBT_vebt_inserti) ) ) ) ).

% vebt_inserti.fixp_induct
tff(fact_7893_tan__pos__pi2__le,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,Xb)) ) ) ).

% tan_pos_pi2_le
tff(fact_7894_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
          & ( aa_real_real(tan_real,X2) = Y ) ) ) ).

% tan_total_pos
tff(fact_7895_tan__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xb)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_7896_tan__mono__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)) ) ) ) ).

% tan_mono_le
tff(fact_7897_tan__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_7898_tan__bound__pi2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,Xb))),one_one_real) ) ).

% tan_bound_pi2
tff(fact_7899_arctan,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_7900_arctan__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,Xb)) = Xb ) ) ) ).

% arctan_tan
tff(fact_7901_arctan__unique,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( ( aa_real_real(tan_real,Xb) = Y )
         => ( aa_real_real(arctan,Y) = Xb ) ) ) ) ).

% arctan_unique
tff(fact_7902_lemma__tan__add1,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))) = aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_7903_tan__diff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(minus_minus_real(Xb),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(minus_minus_real(Xb),Y)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_7904_tan__add,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))),aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_add
tff(fact_7905_tan__total__pi4,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ? [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))),Z2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,Z2) = Xb ) ) ) ).

% tan_total_pi4
tff(fact_7906_disjE__realizer2,axiom,
    ! [P: $o,Q: fun_num_o,Xb: option_num,R4: fun_option_num_o,Fa: option_num,G: fun_num_option_num] :
      ( aa_option_num_o(case_option_o_num((P),Q),Xb)
     => ( ( (P)
         => aa_option_num_o(R4,Fa) )
       => ( ! [Q3: num] :
              ( aa_num_o(Q,Q3)
             => aa_option_num_o(R4,aa_num_option_num(G,Q3)) )
         => aa_option_num_o(R4,aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(Fa,G),Xb)) ) ) ) ).

% disjE_realizer2
tff(fact_7907_disjE__realizer2,axiom,
    ! [P: $o,Q: fun_num_o,Xb: option_num,R4: fun_num_o,Fa: num,G: fun_num_num] :
      ( aa_option_num_o(case_option_o_num((P),Q),Xb)
     => ( ( (P)
         => aa_num_o(R4,Fa) )
       => ( ! [Q3: num] :
              ( aa_num_o(Q,Q3)
             => aa_num_o(R4,aa_num_num(G,Q3)) )
         => aa_num_o(R4,aa_option_num_num(case_option_num_num(Fa,G),Xb)) ) ) ) ).

% disjE_realizer2
tff(fact_7908_disjE__realizer2,axiom,
    ! [P: $o,Q: fun_num_o,Xb: option_num,R4: fun_int_o,Fa: int,G: fun_num_int] :
      ( aa_option_num_o(case_option_o_num((P),Q),Xb)
     => ( ( (P)
         => aa_int_o(R4,Fa) )
       => ( ! [Q3: num] :
              ( aa_num_o(Q,Q3)
             => aa_int_o(R4,aa_num_int(G,Q3)) )
         => aa_int_o(R4,aa_option_num_int(case_option_int_num(Fa,G),Xb)) ) ) ) ).

% disjE_realizer2
tff(fact_7909_disjE__realizer2,axiom,
    ! [P: $o,Q: fun_Pr4796943543331071514_nat_o,Xb: option4927543243414619207at_nat,R4: fun_He8247548780604731983ap_o_o,Fa: heap_Time_Heap_o,G: fun_Pr1329037631615965750Heap_o] :
      ( aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((P),Q),Xb)
     => ( ( (P)
         => aa_Hea4522166450600650424ap_o_o(R4,Fa) )
       => ( ! [Q3: product_prod_nat_nat] :
              ( aa_Pro6150637604445964947_nat_o(Q,Q3)
             => aa_Hea4522166450600650424ap_o_o(R4,aa_Pro691548381575413351Heap_o(G,Q3)) )
         => aa_Hea4522166450600650424ap_o_o(R4,case_o1442776274061689234at_nat(Fa,G,Xb)) ) ) ) ).

% disjE_realizer2
tff(fact_7910_disjE__realizer2,axiom,
    ! [P: $o,Q: fun_Pr4796943543331071514_nat_o,Xb: option4927543243414619207at_nat,R4: fun_o_o,Fa: $o,G: fun_Pr4796943543331071514_nat_o] :
      ( aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((P),Q),Xb)
     => ( ( (P)
         => aa_o_o(R4,(Fa)) )
       => ( ! [Q3: product_prod_nat_nat] :
              ( aa_Pro6150637604445964947_nat_o(Q,Q3)
             => aa_o_o(R4,aa_Pro6150637604445964947_nat_o(G,Q3)) )
         => aa_o_o(R4,aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat((Fa),G),Xb)) ) ) ) ).

% disjE_realizer2
tff(fact_7911_tan__half,axiom,
    ! [Xb: real] : aa_real_real(tan_real,Xb) = aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),Xb))),one_one_real)) ).

% tan_half
tff(fact_7912_exI__realizer,axiom,
    ! [P: fun_num_fun_num_o,Y: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(P,Y),Xb)
     => aa_num_o(aa_num_fun_num_o(P,aa_Pro759708478137009011um_num(product_snd_num_num,aa_num3156681643174609981um_num(product_Pair_num_num(Xb),Y))),aa_Pro759708478137009011um_num(product_fst_num_num,aa_num3156681643174609981um_num(product_Pair_num_num(Xb),Y))) ) ).

% exI_realizer
tff(fact_7913_exI__realizer,axiom,
    ! [P: fun_Pr7973036406301375619_nat_o,Y: produc4813437837504472865T_VEBT,Xb: nat] :
      ( aa_nat_o(aa_Pro2351499262292137340_nat_o(P,Y),Xb)
     => aa_nat_o(aa_Pro2351499262292137340_nat_o(P,aa_Pro3664079603483438877T_VEBT(produc2084898568784432842T_VEBT,aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(Xb),Y))),aa_Pro6878622105917549268BT_nat(produc758997459209783180T_VEBT,aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(Xb),Y))) ) ).

% exI_realizer
tff(fact_7914_exI__realizer,axiom,
    ! [P: fun_num_fun_nat_o,Y: num,Xb: nat] :
      ( aa_nat_o(aa_num_fun_nat_o(P,Y),Xb)
     => aa_nat_o(aa_num_fun_nat_o(P,aa_Pro2818900650166350121um_num(product_snd_nat_num,aa_num5026351239867140595at_num(product_Pair_nat_num(Xb),Y))),aa_Pro6261568864010571487um_nat(product_fst_nat_num,aa_num5026351239867140595at_num(product_Pair_nat_num(Xb),Y))) ) ).

% exI_realizer
tff(fact_7915_exI__realizer,axiom,
    ! [P: fun_nat_fun_nat_o,Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(P,Y),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(P,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y))),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Y))) ) ).

% exI_realizer
tff(fact_7916_exI__realizer,axiom,
    ! [P: fun_int_fun_int_o,Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(P,Y),Xb)
     => aa_int_o(aa_int_fun_int_o(P,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y))),aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Y))) ) ).

% exI_realizer
tff(fact_7917_exI__realizer,axiom,
    ! [P: fun_assn_fun_assn_o,Y: assn,Xb: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(P,Y),Xb)
     => aa_assn_o(aa_assn_fun_assn_o(P,aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(Xb),Y))),aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(Xb),Y))) ) ).

% exI_realizer
tff(fact_7918_conjI__realizer,axiom,
    ! [P: fun_num_o,P2: num,Q: fun_num_o,Q2: num] :
      ( aa_num_o(P,P2)
     => ( aa_num_o(Q,Q2)
       => ( aa_num_o(P,aa_Pro759708478137009011um_num(product_fst_num_num,aa_num3156681643174609981um_num(product_Pair_num_num(P2),Q2)))
          & aa_num_o(Q,aa_Pro759708478137009011um_num(product_snd_num_num,aa_num3156681643174609981um_num(product_Pair_num_num(P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_7919_conjI__realizer,axiom,
    ! [P: fun_nat_o,P2: nat,Q: fun_Pr984851032239314682VEBT_o,Q2: produc4813437837504472865T_VEBT] :
      ( aa_nat_o(P,P2)
     => ( aa_Pro3461009536670225267VEBT_o(Q,Q2)
       => ( aa_nat_o(P,aa_Pro6878622105917549268BT_nat(produc758997459209783180T_VEBT,aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(P2),Q2)))
          & aa_Pro3461009536670225267VEBT_o(Q,aa_Pro3664079603483438877T_VEBT(produc2084898568784432842T_VEBT,aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_7920_conjI__realizer,axiom,
    ! [P: fun_nat_o,P2: nat,Q: fun_num_o,Q2: num] :
      ( aa_nat_o(P,P2)
     => ( aa_num_o(Q,Q2)
       => ( aa_nat_o(P,aa_Pro6261568864010571487um_nat(product_fst_nat_num,aa_num5026351239867140595at_num(product_Pair_nat_num(P2),Q2)))
          & aa_num_o(Q,aa_Pro2818900650166350121um_num(product_snd_nat_num,aa_num5026351239867140595at_num(product_Pair_nat_num(P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_7921_conjI__realizer,axiom,
    ! [P: fun_nat_o,P2: nat,Q: fun_nat_o,Q2: nat] :
      ( aa_nat_o(P,P2)
     => ( aa_nat_o(Q,Q2)
       => ( aa_nat_o(P,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,P2),Q2)))
          & aa_nat_o(Q,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_7922_conjI__realizer,axiom,
    ! [P: fun_int_o,P2: int,Q: fun_int_o,Q2: int] :
      ( aa_int_o(P,P2)
     => ( aa_int_o(Q,Q2)
       => ( aa_int_o(P,aa_Pro6799258434465214121nt_int(product_fst_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2)))
          & aa_int_o(Q,aa_Pro6799258434465214121nt_int(product_snd_int_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_7923_conjI__realizer,axiom,
    ! [P: fun_assn_o,P2: assn,Q: fun_assn_o,Q2: assn] :
      ( aa_assn_o(P,P2)
     => ( aa_assn_o(Q,Q2)
       => ( aa_assn_o(P,aa_Pro2979818503705714077n_assn(produc9167289414957590229n_assn,aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(P2),Q2)))
          & aa_assn_o(Q,aa_Pro2979818503705714077n_assn(produc2051961928117032727n_assn,aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_7924_VEBT__internal_Ovebt__inserti_H_Omono,axiom,
    ! [Xb: produc3960855945107176009Ti_nat] : comple422786390015765981_VEBTi(partia1868168049876374393Ti_nat(heap_T7173139186834293313_VEBTi),heap_T7173139186834293313_VEBTi,aTP_Lamm_zt(Xb)) ).

% VEBT_internal.vebt_inserti'.mono
tff(fact_7925_VEBT__internal_Ovebt__inserti_H_Oraw__induct,axiom,
    ! [P: fun_VE7055638694157540631_nat_o,Xaa: produc3960855945107176009Ti_nat,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit,R: vEBT_VEBTi,N: nat] :
      ( ! [Vebt_inserti2: fun_VE2018681615261037117_VEBTi] :
          ( ! [A5: vEBT_VEBT,B4: vEBT_VEBTi,Ba: nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit,R2: vEBT_VEBTi,N4: nat] :
              ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(Vebt_inserti2,A5),B4),Ba),H3,H4),R2),N4)
             => aa_nat_o(aa_VEB8883426185042993134_nat_o(aa_hea5256493101677210548_nat_o(aa_hea3862133550498874060_nat_o(aa_nat3537556108769025421_nat_o(aa_VEB2168324858650567793_nat_o(aa_VEB4449710241565182864_nat_o(P,A5),B4),Ba),H3),H4),R2),N4) )
         => ! [T: vEBT_VEBT,Ti2: vEBT_VEBTi,X2: nat,Ta: heap_e7401611519738050253t_unit,Tia: heap_e7401611519738050253t_unit,Xa2: vEBT_VEBTi,N3: nat] :
              ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_nat3009484633486537743_VEBTi(aa_VEB600817258366935214_VEBTi(aTP_Lamm_xa(Vebt_inserti2),T),X2),aTP_Lamm_xb(X2)),Ti2),Ta,Tia),Xa2),N3)
             => aa_nat_o(aa_VEB8883426185042993134_nat_o(aa_hea5256493101677210548_nat_o(aa_hea3862133550498874060_nat_o(aa_nat3537556108769025421_nat_o(aa_VEB2168324858650567793_nat_o(aa_VEB4449710241565182864_nat_o(P,T),Ti2),X2),Ta),Tia),Xa2),N3) ) )
     => ( aa_nat_o(aa_VEB8883426185042993134_nat_o(heap_T2071195472996403633_VEBTi(aa_Pro721367001817565044_VEBTi(produc2943724498215716011_VEBTi(produc3258509807451162059_VEBTi(vEBT_V3964819847710782039nserti)),Xaa),H,H2),R),N)
       => aa_nat_o(aa_VEB8883426185042993134_nat_o(aa_hea5256493101677210548_nat_o(aa_hea3862133550498874060_nat_o(produc7249335055465628165_nat_o(produc1140538389304814757_nat_o(P),Xaa),H),H2),R),N) ) ) ).

% VEBT_internal.vebt_inserti'.raw_induct
tff(fact_7926_VEBT__internal_Ovebt__inserti_H_Osimps,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(vEBT_V3964819847710782039nserti,Tb),Tib),Xb) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_nat3009484633486537743_VEBTi(aTP_Lamm_aaq(Tb),Xb),aTP_Lamm_xb(Xb)),Tib) ).

% VEBT_internal.vebt_inserti'.simps
tff(fact_7927_VEBT__internal_Ovebt__buildupi_H_Oelims,axiom,
    ! [Xb: nat,Y: heap_T8145700208782473153_VEBTi] :
      ( ( aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_aau(Va2),half)) ),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_aax(Va2),half)) ) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.elims
tff(fact_7928_VEBT__internal_Ovebt__buildupi_H_Osimps_I3_J,axiom,
    ! [Va: nat] :
      aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_aau(Va),half)) ),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_aax(Va),half)) ) ) ).

% VEBT_internal.vebt_buildupi'.simps(3)
tff(fact_7929_vebt__buildupi_Oelims,axiom,
    ! [Xb: nat,Y: heap_T8145700208782473153_VEBTi] :
      ( ( aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_aaz(Va2),half)) ),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_abb(Va2),half)) ) ) ) ) ) ) ) ).

% vebt_buildupi.elims
tff(fact_7930_vebt__buildupi_Osimps_I3_J,axiom,
    ! [Va: nat] :
      aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_aaz(Va),half)) ),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_abb(Va),half)) ) ) ).

% vebt_buildupi.simps(3)
tff(fact_7931_in__set__enumerate__eq,axiom,
    ! [P2: produc8025551001238799321T_VEBT,N: nat,Xs: list_VEBT_VEBT] :
      ( member8549952807677709168T_VEBT(P2,set_Pr5984661752051438084T_VEBT(enumerate_VEBT_VEBT(N,Xs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),produc8575180428842422559T_VEBT(P2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,produc8575180428842422559T_VEBT(P2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_s6755466524823107622T_VEBT(Xs)),N))
        & ( nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,produc8575180428842422559T_VEBT(P2)),N)) = aa_Pro5446849830759474909T_VEBT(produc8172668247895388509T_VEBT,P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_7932_in__set__enumerate__eq,axiom,
    ! [P2: produc214224863196444774_VEBTi,N: nat,Xs: list_VEBT_VEBTi] :
      ( member763447850065367567_VEBTi(P2,set_Pr4207466110102731387_VEBTi(enumerate_VEBT_VEBTi(N,Xs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),produc8252055991070844170_VEBTi(P2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,produc8252055991070844170_VEBTi(P2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_s7982070591426661849_VEBTi(Xs)),N))
        & ( nth_VEBT_VEBTi(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,produc8252055991070844170_VEBTi(P2)),N)) = produc271786961351840588_VEBTi(P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_7933_in__set__enumerate__eq,axiom,
    ! [P2: produc7716430852924023517t_real,N: nat,Xs: list_real] :
      ( member557208447399453958t_real(P2,set_Pr7149346036329476978t_real(enumerate_real(N,Xs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),product_fst_nat_real(P2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,product_fst_nat_real(P2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_real(Xs)),N))
        & ( aa_nat_real(nth_real(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,product_fst_nat_real(P2)),N)) = product_snd_nat_real(P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_7934_in__set__enumerate__eq,axiom,
    ! [P2: product_prod_nat_o,N: nat,Xs: list_o] :
      ( member6310962623043647828_nat_o(P2,set_Pr1291962091234853352_nat_o(enumerate_o(N,Xs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),product_fst_nat_o(P2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,product_fst_nat_o(P2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_o(Xs)),N))
        & ( aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,product_fst_nat_o(P2)),N))
        <=> product_snd_nat_o(P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_7935_in__set__enumerate__eq,axiom,
    ! [P2: product_prod_nat_int,N: nat,Xs: list_int] :
      ( member4262671552274231302at_int(P2,set_Pr1470767568048878706at_int(enumerate_int(N,Xs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),product_fst_nat_int(P2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,product_fst_nat_int(P2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_int(Xs)),N))
        & ( aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,product_fst_nat_int(P2)),N)) = product_snd_nat_int(P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_7936_in__set__enumerate__eq,axiom,
    ! [P2: product_prod_nat_nat,N: nat,Xs: list_nat] :
      ( member8440522571783428010at_nat(P2,set_Pr5648618587558075414at_nat(enumerate_nat(N,Xs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2))
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_nat(Xs)),N))
        & ( aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,P2)),N)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_7937_length__enumerate,axiom,
    ! [N: nat,Xs: list_real] : size_s7910714270633306959t_real(enumerate_real(N,Xs)) = size_size_list_real(Xs) ).

% length_enumerate
tff(fact_7938_length__enumerate,axiom,
    ! [N: nat,Xs: list_o] : size_s6491369823275344609_nat_o(enumerate_o(N,Xs)) = size_size_list_o(Xs) ).

% length_enumerate
tff(fact_7939_length__enumerate,axiom,
    ! [N: nat,Xs: list_nat] : size_s5460976970255530739at_nat(enumerate_nat(N,Xs)) = size_size_list_nat(Xs) ).

% length_enumerate
tff(fact_7940_length__enumerate,axiom,
    ! [N: nat,Xs: list_int] : size_s2970893825323803983at_int(enumerate_int(N,Xs)) = size_size_list_int(Xs) ).

% length_enumerate
tff(fact_7941_map__snd__enumerate,axiom,
    ! [N: nat,Xs: list_nat] : map_Pr3938374229010428429at_nat(product_snd_nat_nat,enumerate_nat(N,Xs)) = Xs ).

% map_snd_enumerate
tff(fact_7942_TBOUND__of__list,axiom,
    ! [Xs: list_VEBT_VEBTi] : aa_nat_o(time_T6070283812100419266_VEBTi(array_615059503499738533_VEBTi(Xs)),aa_nat_nat(suc,size_s7982070591426661849_VEBTi(Xs))) ).

% TBOUND_of_list
tff(fact_7943_TBOUND__of__list,axiom,
    ! [Xs: list_o] : aa_nat_o(time_TBOUND_array_o(array_of_list_o(Xs)),aa_nat_nat(suc,size_size_list_o(Xs))) ).

% TBOUND_of_list
tff(fact_7944_TBOUND__of__list,axiom,
    ! [Xs: list_nat] : aa_nat_o(time_T3621710982123377501ay_nat(array_of_list_nat(Xs)),aa_nat_nat(suc,size_size_list_nat(Xs))) ).

% TBOUND_of_list
tff(fact_7945_TBOUND__of__list,axiom,
    ! [Xs: list_int] : aa_nat_o(time_T8667231999468956601ay_int(array_of_list_int(Xs)),aa_nat_nat(suc,size_size_list_int(Xs))) ).

% TBOUND_of_list
tff(fact_7946_time__array__of__list,axiom,
    ! [Xs: list_VEBT_VEBTi,H: heap_e7401611519738050253t_unit] : time_t9122064381910598399_VEBTi(array_615059503499738533_VEBTi(Xs),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_s7982070591426661849_VEBTi(Xs)),one_one_nat) ).

% time_array_of_list
tff(fact_7947_time__array__of__list,axiom,
    ! [Xs: list_o,H: heap_e7401611519738050253t_unit] : time_time_array_o(array_of_list_o(Xs),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_o(Xs)),one_one_nat) ).

% time_array_of_list
tff(fact_7948_time__array__of__list,axiom,
    ! [Xs: list_nat,H: heap_e7401611519738050253t_unit] : time_time_array_nat(array_of_list_nat(Xs),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_nat(Xs)),one_one_nat) ).

% time_array_of_list
tff(fact_7949_time__array__of__list,axiom,
    ! [Xs: list_int,H: heap_e7401611519738050253t_unit] : time_time_array_int(array_of_list_int(Xs),H) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_int(Xs)),one_one_nat) ).

% time_array_of_list
tff(fact_7950_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_Pr744662078594809490T_VEBT(enumerate_VEBT_VEBT(N,Xs),M) = produc599794634098209291T_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M),nth_VEBT_VEBT(Xs,M)) ) ) ).

% nth_enumerate_eq
tff(fact_7951_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBTi,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_Pr3244165891152107629_VEBTi(enumerate_VEBT_VEBTi(N,Xs),M) = produc2649746096677893406_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M),nth_VEBT_VEBTi(Xs,M)) ) ) ).

% nth_enumerate_eq
tff(fact_7952_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_P6730324909620535719T_VEBT,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_s4764337671732037139T_VEBT(Xs))
     => ( nth_Pr5469784954002723455T_VEBT(enumer7493320076530018694T_VEBT(N,Xs),M) = aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),nth_Pr2419613052044807976T_VEBT(Xs,M)) ) ) ).

% nth_enumerate_eq
tff(fact_7953_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_num,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_num(Xs))
     => ( nth_Pr8326237132889035090at_num(enumerate_num(N,Xs),M) = aa_num5026351239867140595at_num(product_Pair_nat_num(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),aa_nat_num(nth_num(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_7954_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_real,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_real(Xs))
     => ( nth_Pr7767817059697521252t_real(enumerate_real(N,Xs),M) = produc7837566107596912789t_real(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M),aa_nat_real(nth_real(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_7955_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_o,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_o(Xs))
     => ( nth_Pr112076138515278198_nat_o(enumerate_o(N,Xs),M) = product_Pair_nat_o(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M),aa_nat_o(nth_o(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_7956_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_nat(Xs))
     => ( aa_nat7603221163087113075at_nat(nth_Pr7617993195940197384at_nat(enumerate_nat(N,Xs)),M) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),aa_nat_nat(nth_nat(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_7957_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_int,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_int(Xs))
     => ( nth_Pr3440142176431000676at_int(enumerate_int(N,Xs),M) = product_Pair_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M),aa_nat_int(nth_int(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_7958_VEBT__internal_Ovebt__buildupi_H_Opelims,axiom,
    ! [Xb: nat,Y: heap_T8145700208782473153_VEBTi] :
      ( ( aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V254170901696579886pi_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) )
             => ~ aa_nat_o(accp_nat(vEBT_V254170901696579886pi_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) )
               => ~ aa_nat_o(accp_nat(vEBT_V254170901696579886pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_aau(Va2),half)) ),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_aax(Va2),half)) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_V254170901696579886pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.pelims
tff(fact_7959_VEBTi_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] : aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X12,X13),X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_a6397454172108246045_VEBTi(size_size_VEBT_VEBTi,X13)),aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBTi.size(3)
tff(fact_7960_vebt__buildupi_Opelims,axiom,
    ! [Xb: nat,Y: heap_T8145700208782473153_VEBTi] :
      ( ( aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_v1230518104690509829pi_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) )
             => ~ aa_nat_o(accp_nat(vEBT_v1230518104690509829pi_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($false,$false)) )
               => ~ aa_nat_o(accp_nat(vEBT_v1230518104690509829pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_aaz(Va2),half)) ),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            heap_T5877712393672139267_VEBTi(vEBT_V1859673955506687831_VEBTi(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,half)),aa_nat4125268785154830727_VEBTi(aTP_Lamm_abb(Va2),half)) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_v1230518104690509829pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildupi.pelims
tff(fact_7961_sin__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
     => ( aa_real_real(sin_real,Xb) = aa_real_real(divide_divide_real(aa_real_real(tan_real,Xb)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_7962_cos__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
     => ( aa_real_real(cos_real,Xb) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_7963_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sqrt,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_7964_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_7965_real__sqrt__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% real_sqrt_less_iff
tff(fact_7966_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_7967_real__sqrt__lt__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% real_sqrt_lt_0_iff
tff(fact_7968_real__sqrt__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% real_sqrt_le_0_iff
tff(fact_7969_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_7970_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ).

% real_sqrt_gt_1_iff
tff(fact_7971_real__sqrt__lt__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ).

% real_sqrt_lt_1_iff
tff(fact_7972_real__sqrt__four,axiom,
    aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))) = numeral_numeral_real(aa_num_num(bit0,one)) ).

% real_sqrt_four
tff(fact_7973_real__sqrt__abs,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xb) ).

% real_sqrt_abs
tff(fact_7974_real__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb ) ) ).

% real_sqrt_pow2
tff(fact_7975_real__sqrt__pow2__iff,axiom,
    ! [Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% real_sqrt_pow2_iff
tff(fact_7976_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [Xb: real,Y: real,Xaa: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_7977_real__sqrt__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_7978_real__sqrt__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_gt_zero
tff(fact_7979_real__sqrt__eq__zero__cancel,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(sqrt,Xb) = zero_zero_real )
       => ( Xb = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_7980_real__sqrt__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_ge_zero
tff(fact_7981_real__div__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(divide_divide_real(Xb),aa_real_real(sqrt,Xb)) = aa_real_real(sqrt,Xb) ) ) ).

% real_div_sqrt
tff(fact_7982_sqrt__add__le__add__sqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_7983_sqrt2__less__2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sqrt2_less_2
tff(fact_7984_real__less__rsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_7985_sqrt__le__D,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% sqrt_le_D
tff(fact_7986_real__le__rsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_7987_real__le__lsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_7988_real__sqrt__unique,axiom,
    ! [Y: real,Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(sqrt,Xb) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_7989_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),U)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_7990_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Xb )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_7991_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Y )
     => ( Xb = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_7992_real__sqrt__sum__squares__ge1,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_7993_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_7994_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C: real,B2: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_7995_sqrt__ge__absD,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(sqrt,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y) ) ).

% sqrt_ge_absD
tff(fact_7996_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_45
tff(fact_7997_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_45
tff(fact_7998_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one))) ).

% tan_60
tff(fact_7999_real__less__lsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_8000_sqrt__sum__squares__le__sum,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_8001_sqrt__even__pow2,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% sqrt_even_pow2
tff(fact_8002_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_8003_real__sqrt__ge__abs2,axiom,
    ! [Y: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs2
tff(fact_8004_real__sqrt__ge__abs1,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs1
tff(fact_8005_ln__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,Xb)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Xb)),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_8006_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% cos_30
tff(fact_8007_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit1,one)))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% sin_60
tff(fact_8008_arsinh__real__def,axiom,
    ! [Xb: real] : aa_real_real(arsinh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_8009_real__sqrt__power__even,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),N) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_8010_arsinh__real__aux,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_8011_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xb: real,Y: real,Xaa: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(times_times_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_8012_arith__geo__mean__sqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(times_times_real(Xb),Y))),aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_8013_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_8014_cos__x__y__le__one,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_8015_real__sqrt__sum__squares__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(divide_divide_real(U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(divide_divide_real(U),aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_8016_arcosh__real__def,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_real(arcosh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_8017_cos__arctan,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(arctan,Xb)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_8018_sin__arctan,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(arctan,Xb)) = aa_real_real(divide_divide_real(Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_8019_sqrt__sum__squares__half__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(U),numeral_numeral_real(aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(divide_divide_real(U),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_8020_sin__cos__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xb))
     => ( aa_real_real(sin_real,Xb) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_8021_arctan__half,axiom,
    ! [Xb: real] : aa_real_real(arctan,Xb) = aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(divide_divide_real(Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_8022_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [Xb: nat,Y: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = one_one_int )
             => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = one_one_int )
               => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N3: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N3))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
tff(fact_8023_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,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
             => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
               => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $let(
                              half: nat,
                              half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
tff(fact_8024_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,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
             => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $let(
                              half: nat,
                              half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
tff(fact_8025_VEBT__internal_OTb_Opelims,axiom,
    ! [Xb: nat,Y: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) )
             => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) )
               => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N3: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,aa_nat_nat(suc,N3))) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
tff(fact_8026_VEBT__internal_OTb_H_Opelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
             => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N3: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,aa_nat_nat(suc,N3))) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
tff(fact_8027_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = aa_nat_nat(suc,zero_zero_nat) )
             => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = aa_nat_nat(suc,zero_zero_nat) )
               => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N3: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,N3)) )
                 => ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N3),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N3))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_8028_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] : aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X12,X13),X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_a6397454172108246045_VEBTi(vEBT_size_VEBTi,X13)),aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBTi.size_gen(1)
tff(fact_8029_obtain__set__succ,axiom,
    ! [Xb: nat,Z: nat,A3: set_nat,B5: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(A3),Z)
       => ( aa_set_nat_o(finite_finite_nat,B5)
         => ( ( A3 = B5 )
           => ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(A3),Xb),X_1) ) ) ) ) ).

% obtain_set_succ
tff(fact_8030_obtain__set__pred,axiom,
    ! [Z: nat,Xb: nat,A3: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xb)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(A3),Z)
       => ( aa_set_nat_o(finite_finite_nat,A3)
         => ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(A3),Xb),X_1) ) ) ) ).

% obtain_set_pred
tff(fact_8031_bezw_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Xaa)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_8032_set__vebt__finite,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(Tb)) ) ).

% set_vebt_finite
tff(fact_8033_pred__none__empty,axiom,
    ! [Xs: set_nat,A2: nat] :
      ( ~ ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),A2),X_1)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),A2) ) ) ) ).

% pred_none_empty
tff(fact_8034_succ__none__empty,axiom,
    ! [Xs: set_nat,A2: nat] :
      ( ~ ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),A2),X_1)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),X3) ) ) ) ).

% succ_none_empty
tff(fact_8035_infinite__Icc__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A2,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% infinite_Icc_iff
tff(fact_8036_infinite__Icc__iff,axiom,
    ! [A2: real,B2: real] :
      ( ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A2,B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% infinite_Icc_iff
tff(fact_8037_summable__If__finite__set,axiom,
    ! [A3: set_nat,Fa: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_abc(A3),Fa)) ) ).

% summable_If_finite_set
tff(fact_8038_summable__If__finite__set,axiom,
    ! [A3: set_nat,Fa: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_abd(A3),Fa)) ) ).

% summable_If_finite_set
tff(fact_8039_summable__If__finite__set,axiom,
    ! [A3: set_nat,Fa: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_abe(A3),Fa)) ) ).

% summable_If_finite_set
tff(fact_8040_summable__If__finite__set,axiom,
    ! [A3: set_nat,Fa: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_abf(A3),Fa)) ) ).

% summable_If_finite_set
tff(fact_8041_summable__If__finite,axiom,
    ! [P: fun_nat_o,Fa: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_abg(P),Fa)) ) ).

% summable_If_finite
tff(fact_8042_summable__If__finite,axiom,
    ! [P: fun_nat_o,Fa: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_abh(P),Fa)) ) ).

% summable_If_finite
tff(fact_8043_summable__If__finite,axiom,
    ! [P: fun_nat_o,Fa: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_abi(P),Fa)) ) ).

% summable_If_finite
tff(fact_8044_summable__If__finite,axiom,
    ! [P: fun_nat_o,Fa: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_abj(P),Fa)) ) ).

% summable_If_finite
tff(fact_8045_finite__nat__set__iff__bounded,axiom,
    ! [N2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N2)
    <=> ? [M6: nat] :
        ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),M6) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_8046_bounded__nat__set__is__finite,axiom,
    ! [N2: set_nat,N: nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),N) )
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% bounded_nat_set_is_finite
tff(fact_8047_finite__nat__set__iff__bounded__le,axiom,
    ! [N2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N2)
    <=> ? [M6: nat] :
        ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),M6) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_8048_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_int,S4: set_int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => finite6197958912794628473et_int(collect_set_int(aa_set5425576183772390127_int_o(aTP_Lamm_abk(S3),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_8049_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_complex,S4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => finite6551019134538273531omplex(collect_set_complex(aa_set8643480048548153067plex_o(aTP_Lamm_abl(S3),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_8050_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_Code_integer,S4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => finite6931041176100689706nteger(collec574505750873337192nteger(aa_set4044230130517860621eger_o(aTP_Lamm_abm(S3),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_8051_finite__if__eq__beyond__finite,axiom,
    ! [S3: set_nat,S4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => finite1152437895449049373et_nat(collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_abn(S3),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_8052_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_o,I: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_abo(P),I))) ).

% finite_M_bounded_by_nat
tff(fact_8053_finite__less__ub,axiom,
    ! [Fa: fun_nat_nat,U: nat] :
      ( ! [N3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N3),aa_nat_nat(Fa,N3))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_abp(Fa),U))) ) ).

% finite_less_ub
tff(fact_8054_finite__lists__length__eq,axiom,
    ! [A3: set_VEBT_VEBT,N: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_abq(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_8055_finite__lists__length__eq,axiom,
    ! [A3: set_Pr5949110396991348497n_assn,N: nat] :
      ( finite5137929494490007386n_assn(A3)
     => finite1351478129840809056n_assn(collec8177951099088521122n_assn(aa_nat2893684320935325170assn_o(aTP_Lamm_abr(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_8056_finite__lists__length__eq,axiom,
    ! [A3: set_complex,N: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_abs(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_8057_finite__lists__length__eq,axiom,
    ! [A3: set_Code_integer,N: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aa_nat2697030861539126eger_o(aTP_Lamm_abt(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_8058_finite__lists__length__eq,axiom,
    ! [A3: set_real,N: nat] :
      ( aa_set_real_o(finite_finite_real,A3)
     => finite306553202115118035t_real(collect_list_real(aa_nat3629845878562903719real_o(aTP_Lamm_abu(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_8059_finite__lists__length__eq,axiom,
    ! [A3: set_o,N: nat] :
      ( finite_finite_o(A3)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_abv(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_8060_finite__lists__length__eq,axiom,
    ! [A3: set_int,N: nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_abw(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_8061_finite__lists__length__eq,axiom,
    ! [A3: set_nat,N: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_abx(A3),N))) ) ).

% finite_lists_length_eq
tff(fact_8062_infinite__Icc,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A2,B2)) ) ).

% infinite_Icc
tff(fact_8063_infinite__Icc,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A2,B2)) ) ).

% infinite_Icc
tff(fact_8064_finite__lists__length__le,axiom,
    ! [A3: set_VEBT_VEBT,N: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_aby(A3),N))) ) ).

% finite_lists_length_le
tff(fact_8065_finite__lists__length__le,axiom,
    ! [A3: set_Pr5949110396991348497n_assn,N: nat] :
      ( finite5137929494490007386n_assn(A3)
     => finite1351478129840809056n_assn(collec8177951099088521122n_assn(aa_nat2893684320935325170assn_o(aTP_Lamm_abz(A3),N))) ) ).

% finite_lists_length_le
tff(fact_8066_finite__lists__length__le,axiom,
    ! [A3: set_complex,N: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_aca(A3),N))) ) ).

% finite_lists_length_le
tff(fact_8067_finite__lists__length__le,axiom,
    ! [A3: set_Code_integer,N: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aa_nat2697030861539126eger_o(aTP_Lamm_acb(A3),N))) ) ).

% finite_lists_length_le
tff(fact_8068_finite__lists__length__le,axiom,
    ! [A3: set_real,N: nat] :
      ( aa_set_real_o(finite_finite_real,A3)
     => finite306553202115118035t_real(collect_list_real(aa_nat3629845878562903719real_o(aTP_Lamm_acc(A3),N))) ) ).

% finite_lists_length_le
tff(fact_8069_finite__lists__length__le,axiom,
    ! [A3: set_o,N: nat] :
      ( finite_finite_o(A3)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_acd(A3),N))) ) ).

% finite_lists_length_le
tff(fact_8070_finite__lists__length__le,axiom,
    ! [A3: set_int,N: nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_ace(A3),N))) ) ).

% finite_lists_length_le
tff(fact_8071_finite__lists__length__le,axiom,
    ! [A3: set_nat,N: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_acf(A3),N))) ) ).

% finite_lists_length_le
tff(fact_8072_summable__finite,axiom,
    ! [N2: set_nat,Fa: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,N2)
     => ( ! [N3: nat] :
            ( ~ aa_set_nat_o(member_nat(N3),N2)
           => ( aa_nat_complex(Fa,N3) = zero_zero_complex ) )
       => summable_complex(Fa) ) ) ).

% summable_finite
tff(fact_8073_summable__finite,axiom,
    ! [N2: set_nat,Fa: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,N2)
     => ( ! [N3: nat] :
            ( ~ aa_set_nat_o(member_nat(N3),N2)
           => ( aa_nat_real(Fa,N3) = zero_zero_real ) )
       => summable_real(Fa) ) ) ).

% summable_finite
tff(fact_8074_summable__finite,axiom,
    ! [N2: set_nat,Fa: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,N2)
     => ( ! [N3: nat] :
            ( ~ aa_set_nat_o(member_nat(N3),N2)
           => ( aa_nat_nat(Fa,N3) = zero_zero_nat ) )
       => summable_nat(Fa) ) ) ).

% summable_finite
tff(fact_8075_summable__finite,axiom,
    ! [N2: set_nat,Fa: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,N2)
     => ( ! [N3: nat] :
            ( ~ aa_set_nat_o(member_nat(N3),N2)
           => ( aa_nat_int(Fa,N3) = zero_zero_int ) )
       => summable_int(Fa) ) ) ).

% summable_finite
tff(fact_8076_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_nc,M))) ) ).

% finite_divisors_nat
tff(fact_8077_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N2: set_nat,N: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_8078_finite__roots__unity,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),N)
     => aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aTP_Lamm_acg,N))) ) ).

% finite_roots_unity
tff(fact_8079_finite__roots__unity,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),N)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_ach,N))) ) ).

% finite_roots_unity
tff(fact_8080_VEBTi_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,vEBT_Leafi((X21),(X222))) = zero_zero_nat ).

% VEBTi.size_gen(2)
tff(fact_8081_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aci,K))) ).

% finite_Collect_le_nat
tff(fact_8082_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fg,K))) ).

% finite_Collect_less_nat
tff(fact_8083_finite__Collect__subsets,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_acj(A3))) ) ).

% finite_Collect_subsets
tff(fact_8084_finite__Collect__subsets,axiom,
    ! [A3: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_ack(A3))) ) ).

% finite_Collect_subsets
tff(fact_8085_finite__Collect__subsets,axiom,
    ! [A3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => finite6931041176100689706nteger(collec574505750873337192nteger(aTP_Lamm_acl(A3))) ) ).

% finite_Collect_subsets
tff(fact_8086_finite__Collect__subsets,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => finite1152437895449049373et_nat(collect_set_nat(aTP_Lamm_acm(A3))) ) ).

% finite_Collect_subsets
tff(fact_8087_finite__induct__select,axiom,
    ! [S3: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [T3: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(T3),S3)
             => ( aa_set_VEBT_VEBT_o(P,T3)
               => ? [X3: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),minus_5127226145743854075T_VEBT(S3,T3))
                    & aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X3,T3)) ) ) )
         => aa_set_VEBT_VEBT_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_8088_finite__induct__select,axiom,
    ! [S3: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [T3: set_complex] :
              ( aa_set_complex_o(ord_less_set_complex(T3),S3)
             => ( aa_set_complex_o(P,T3)
               => ? [X3: complex] :
                    ( aa_set_complex_o(member_complex(X3),minus_811609699411566653omplex(S3,T3))
                    & aa_set_complex_o(P,insert_complex(X3,T3)) ) ) )
         => aa_set_complex_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_8089_finite__induct__select,axiom,
    ! [S3: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [T3: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(T3),S3)
             => ( aa_set5643387332196575111eger_o(P,T3)
               => ? [X3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X3),minus_2355218937544613996nteger(S3,T3))
                    & aa_set5643387332196575111eger_o(P,insert_Code_integer(X3,T3)) ) ) )
         => aa_set5643387332196575111eger_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_8090_finite__induct__select,axiom,
    ! [S3: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [T3: set_real] :
              ( aa_set_real_o(ord_less_set_real(T3),S3)
             => ( aa_set_real_o(P,T3)
               => ? [X3: real] :
                    ( aa_set_real_o(member_real(X3),minus_minus_set_real(S3,T3))
                    & aa_set_real_o(P,insert_real(X3,T3)) ) ) )
         => aa_set_real_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_8091_finite__induct__select,axiom,
    ! [S3: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [T3: set_int] :
              ( aa_set_int_o(ord_less_set_int(T3),S3)
             => ( aa_set_int_o(P,T3)
               => ? [X3: int] :
                    ( aa_set_int_o(member_int(X3),minus_minus_set_int(S3,T3))
                    & aa_set_int_o(P,insert_int(X3,T3)) ) ) )
         => aa_set_int_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_8092_finite__induct__select,axiom,
    ! [S3: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [T3: set_nat] :
              ( aa_set_nat_o(ord_less_set_nat(T3),S3)
             => ( aa_set_nat_o(P,T3)
               => ? [X3: nat] :
                    ( aa_set_nat_o(member_nat(X3),minus_minus_set_nat(S3,T3))
                    & aa_set_nat_o(P,insert_nat(X3,T3)) ) ) )
         => aa_set_nat_o(P,S3) ) ) ) ).

% finite_induct_select
tff(fact_8093_set__encode__insert,axiom,
    ! [A3: set_nat,N: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ~ aa_set_nat_o(member_nat(N),A3)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(N,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_set_nat_nat(nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_8094_finite__Collect__disjI,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_acn(P),Q)))
    <=> ( finite2998713641127702882nt_int(collec213857154873943460nt_int(P))
        & finite2998713641127702882nt_int(collec213857154873943460nt_int(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_8095_finite__Collect__disjI,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aco(P),Q)))
    <=> ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
        & aa_set_nat_o(finite_finite_nat,collect_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_8096_finite__Collect__disjI,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_acp(P),Q)))
    <=> ( aa_set_int_o(finite_finite_int,collect_int(P))
        & aa_set_int_o(finite_finite_int,collect_int(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_8097_finite__Collect__disjI,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_acq(P),Q)))
    <=> ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
        & aa_set_complex_o(finite3207457112153483333omplex,collect_complex(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_8098_finite__Collect__disjI,axiom,
    ! [P: fun_Code_integer_o,Q: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_acr(P),Q)))
    <=> ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(P))
        & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_8099_finite__Collect__conjI,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ( finite2998713641127702882nt_int(collec213857154873943460nt_int(P))
        | finite2998713641127702882nt_int(collec213857154873943460nt_int(Q)) )
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_acs(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_8100_finite__Collect__conjI,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
        | aa_set_nat_o(finite_finite_nat,collect_nat(Q)) )
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_act(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_8101_finite__Collect__conjI,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ( aa_set_int_o(finite_finite_int,collect_int(P))
        | aa_set_int_o(finite_finite_int,collect_int(Q)) )
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_acu(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_8102_finite__Collect__conjI,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
        | aa_set_complex_o(finite3207457112153483333omplex,collect_complex(Q)) )
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_acv(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_8103_finite__Collect__conjI,axiom,
    ! [P: fun_Code_integer_o,Q: fun_Code_integer_o] :
      ( ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(P))
        | aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(Q)) )
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_acw(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_8104_finite__interval__int1,axiom,
    ! [A2: int,B2: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_acx(A2),B2))) ).

% finite_interval_int1
tff(fact_8105_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_acy(A2),B2))) ).

% finite_interval_int4
tff(fact_8106_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_acz(N),C))) ) ).

% finite_nth_roots
tff(fact_8107_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ada(A2),B2))) ).

% finite_interval_int3
tff(fact_8108_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_adb(A2),B2))) ).

% finite_interval_int2
tff(fact_8109_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_8110_finite__maxlen,axiom,
    ! [M8: set_list_real] :
      ( finite306553202115118035t_real(M8)
     => ? [N3: nat] :
        ! [X3: list_real] :
          ( member_list_real(X3,M8)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_real(X3)),N3) ) ) ).

% finite_maxlen
tff(fact_8111_finite__maxlen,axiom,
    ! [M8: set_list_o] :
      ( finite_finite_list_o(M8)
     => ? [N3: nat] :
        ! [X3: list_o] :
          ( member_list_o(X3,M8)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(X3)),N3) ) ) ).

% finite_maxlen
tff(fact_8112_finite__maxlen,axiom,
    ! [M8: set_list_nat] :
      ( finite8100373058378681591st_nat(M8)
     => ? [N3: nat] :
        ! [X3: list_nat] :
          ( member_list_nat(X3,M8)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(X3)),N3) ) ) ).

% finite_maxlen
tff(fact_8113_finite__maxlen,axiom,
    ! [M8: set_list_int] :
      ( finite3922522038869484883st_int(M8)
     => ? [N3: nat] :
        ! [X3: list_int] :
          ( member_list_int(X3,M8)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(X3)),N3) ) ) ).

% finite_maxlen
tff(fact_8114_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_my,I))) ) ).

% finite_divisors_int
tff(fact_8115_set__encode__inf,axiom,
    ! [A3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_nat(nat_set_encode,A3) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_8116_pigeonhole__infinite__rel,axiom,
    ! [A3: set_VEBT_VEBT,B5: set_nat,R4: fun_VE6480751511521678896_nat_o] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
     => ( aa_set_nat_o(finite_finite_nat,B5)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A3)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B5)
                  & aa_nat_o(aa_VEB2314620471664397097_nat_o(R4,X2),Xa) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),B5)
              & ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_nat1925708948643195573VEBT_o(aa_fun2405540735884763746VEBT_o(aTP_Lamm_adc(A3),R4),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8117_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B5: set_nat,R4: fun_real_fun_nat_o] :
      ( ~ aa_set_real_o(finite_finite_real,A3)
     => ( aa_set_nat_o(finite_finite_nat,B5)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A3)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B5)
                  & aa_nat_o(aa_real_fun_nat_o(R4,X2),Xa) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),B5)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_add(A3),R4),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8118_pigeonhole__infinite__rel,axiom,
    ! [A3: set_VEBT_VEBT,B5: set_int,R4: fun_VE4235606543851232980_int_o] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
     => ( aa_set_int_o(finite_finite_int,B5)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A3)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B5)
                  & aa_int_o(aa_VEB493578126231709133_int_o(R4,X2),Xa) ) )
         => ? [X2: int] :
              ( aa_set_int_o(member_int(X2),B5)
              & ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_int8839263053405020433VEBT_o(aa_fun8807180201667926370VEBT_o(aTP_Lamm_ade(A3),R4),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8119_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B5: set_int,R4: fun_real_fun_int_o] :
      ( ~ aa_set_real_o(finite_finite_real,A3)
     => ( aa_set_int_o(finite_finite_int,B5)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A3)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B5)
                  & aa_int_o(aa_real_fun_int_o(R4,X2),Xa) ) )
         => ? [X2: int] :
              ( aa_set_int_o(member_int(X2),B5)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_adf(A3),R4),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8120_pigeonhole__infinite__rel,axiom,
    ! [A3: set_VEBT_VEBT,B5: set_complex,R4: fun_VE7631951351884037842plex_o] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B5)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A3)
             => ? [Xa: complex] :
                  ( aa_set_complex_o(member_complex(Xa),B5)
                  & aa_complex_o(aa_VEB8036846806599048395plex_o(R4,X2),Xa) ) )
         => ? [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),B5)
              & ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_com3651963455866140051VEBT_o(aa_fun7268111043052431330VEBT_o(aTP_Lamm_adg(A3),R4),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8121_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B5: set_complex,R4: fun_re7319081385389557872plex_o] :
      ( ~ aa_set_real_o(finite_finite_real,A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B5)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A3)
             => ? [Xa: complex] :
                  ( aa_set_complex_o(member_complex(Xa),B5)
                  & aa_complex_o(aa_rea2382570592068293209plex_o(R4,X2),Xa) ) )
         => ? [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),B5)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_adh(A3),R4),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8122_pigeonhole__infinite__rel,axiom,
    ! [A3: set_VEBT_VEBT,B5: set_Code_integer,R4: fun_VE1909044406494083811eger_o] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B5)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A3)
             => ? [Xa: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),B5)
                  & aa_Code_integer_o(aa_VEB1409644639000594780eger_o(R4,X2),Xa) ) )
         => ? [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),B5)
              & ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_Cod5040069908416755522VEBT_o(aa_fun1637734041313256866VEBT_o(aTP_Lamm_adi(A3),R4),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8123_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B5: set_Code_integer,R4: fun_re3424108668441675905eger_o] :
      ( ~ aa_set_real_o(finite_finite_real,A3)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B5)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A3)
             => ? [Xa: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),B5)
                  & aa_Code_integer_o(aa_rea4090721512283939562eger_o(R4,X2),Xa) ) )
         => ? [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),B5)
              & ~ aa_set_real_o(finite_finite_real,collect_real(aa_Cod1315720040876898922real_o(aa_fun4247632250071955964real_o(aTP_Lamm_adj(A3),R4),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8124_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B5: set_nat,R4: fun_nat_fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B5)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A3)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B5)
                  & aa_nat_o(aa_nat_fun_nat_o(R4,X2),Xa) ) )
         => ? [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),B5)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_adk(A3),R4),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8125_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B5: set_int,R4: fun_nat_fun_int_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_int_o(finite_finite_int,B5)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A3)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B5)
                  & aa_int_o(aa_nat_fun_int_o(R4,X2),Xa) ) )
         => ? [X2: int] :
              ( aa_set_int_o(member_int(X2),B5)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_adl(A3),R4),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_8126_not__finite__existsD,axiom,
    ! [P: fun_Pr3238374608701176162_int_o] :
      ( ~ finite2998713641127702882nt_int(collec213857154873943460nt_int(P))
     => ? [X_1: product_prod_int_int] : aa_Pro4592068669816069595_int_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_8127_not__finite__existsD,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ? [X_1: nat] : aa_nat_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_8128_not__finite__existsD,axiom,
    ! [P: fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,collect_int(P))
     => ? [X_1: int] : aa_int_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_8129_not__finite__existsD,axiom,
    ! [P: fun_complex_o] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
     => ? [X_1: complex] : aa_complex_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_8130_not__finite__existsD,axiom,
    ! [P: fun_Code_integer_o] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(P))
     => ? [X_1: code_integer] : aa_Code_integer_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_8131_finite__has__maximal2,axiom,
    ! [A3: set_real,A2: real] :
      ( aa_set_real_o(finite_finite_real,A3)
     => ( aa_set_real_o(member_real(A2),A3)
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),A3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X2)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8132_finite__has__maximal2,axiom,
    ! [A3: set_Code_integer,A2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A2),A3)
       => ? [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A3)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),X2)
            & ! [Xa: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A3)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8133_finite__has__maximal2,axiom,
    ! [A3: set_set_nat,A2: set_nat] :
      ( finite1152437895449049373et_nat(A3)
     => ( member_set_nat(A2,A3)
       => ? [X2: set_nat] :
            ( member_set_nat(X2,A3)
            & aa_set_nat_o(ord_less_eq_set_nat(A2),X2)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A3)
               => ( aa_set_nat_o(ord_less_eq_set_nat(X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8134_finite__has__maximal2,axiom,
    ! [A3: set_rat,A2: rat] :
      ( aa_set_rat_o(finite_finite_rat,A3)
     => ( aa_set_rat_o(member_rat(A2),A3)
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A3)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),X2)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A3)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8135_finite__has__maximal2,axiom,
    ! [A3: set_num,A2: num] :
      ( aa_set_num_o(finite_finite_num,A3)
     => ( aa_set_num_o(member_num(A2),A3)
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),A3)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),X2)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A3)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8136_finite__has__maximal2,axiom,
    ! [A3: set_nat,A2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(member_nat(A2),A3)
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),X2)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8137_finite__has__maximal2,axiom,
    ! [A3: set_int,A2: int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(member_int(A2),A3)
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A3)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),X2)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A3)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8138_finite__has__minimal2,axiom,
    ! [A3: set_real,A2: real] :
      ( aa_set_real_o(finite_finite_real,A3)
     => ( aa_set_real_o(member_real(A2),A3)
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),A3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),A2)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_8139_finite__has__minimal2,axiom,
    ! [A3: set_Code_integer,A2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A2),A3)
       => ? [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A3)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),A2)
            & ! [Xa: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A3)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_8140_finite__has__minimal2,axiom,
    ! [A3: set_set_nat,A2: set_nat] :
      ( finite1152437895449049373et_nat(A3)
     => ( member_set_nat(A2,A3)
       => ? [X2: set_nat] :
            ( member_set_nat(X2,A3)
            & aa_set_nat_o(ord_less_eq_set_nat(X2),A2)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A3)
               => ( aa_set_nat_o(ord_less_eq_set_nat(Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_8141_finite__has__minimal2,axiom,
    ! [A3: set_rat,A2: rat] :
      ( aa_set_rat_o(finite_finite_rat,A3)
     => ( aa_set_rat_o(member_rat(A2),A3)
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A3)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),A2)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A3)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_8142_finite__has__minimal2,axiom,
    ! [A3: set_num,A2: num] :
      ( aa_set_num_o(finite_finite_num,A3)
     => ( aa_set_num_o(member_num(A2),A3)
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),A3)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),A2)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A3)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_8143_finite__has__minimal2,axiom,
    ! [A3: set_nat,A2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(member_nat(A2),A3)
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),A2)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_8144_finite__has__minimal2,axiom,
    ! [A3: set_int,A2: int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(member_int(A2),A3)
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A3)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),A2)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A3)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_8145_finite__psubset__induct,axiom,
    ! [A3: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [A7: set_nat] :
            ( aa_set_nat_o(finite_finite_nat,A7)
           => ( ! [B7: set_nat] :
                  ( aa_set_nat_o(ord_less_set_nat(B7),A7)
                 => aa_set_nat_o(P,B7) )
             => aa_set_nat_o(P,A7) ) )
       => aa_set_nat_o(P,A3) ) ) ).

% finite_psubset_induct
tff(fact_8146_finite__psubset__induct,axiom,
    ! [A3: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [A7: set_int] :
            ( aa_set_int_o(finite_finite_int,A7)
           => ( ! [B7: set_int] :
                  ( aa_set_int_o(ord_less_set_int(B7),A7)
                 => aa_set_int_o(P,B7) )
             => aa_set_int_o(P,A7) ) )
       => aa_set_int_o(P,A3) ) ) ).

% finite_psubset_induct
tff(fact_8147_finite__psubset__induct,axiom,
    ! [A3: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [A7: set_complex] :
            ( aa_set_complex_o(finite3207457112153483333omplex,A7)
           => ( ! [B7: set_complex] :
                  ( aa_set_complex_o(ord_less_set_complex(B7),A7)
                 => aa_set_complex_o(P,B7) )
             => aa_set_complex_o(P,A7) ) )
       => aa_set_complex_o(P,A3) ) ) ).

% finite_psubset_induct
tff(fact_8148_finite__psubset__induct,axiom,
    ! [A3: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ! [A7: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
           => ( ! [B7: set_Code_integer] :
                  ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(B7),A7)
                 => aa_set5643387332196575111eger_o(P,B7) )
             => aa_set5643387332196575111eger_o(P,A7) ) )
       => aa_set5643387332196575111eger_o(P,A3) ) ) ).

% finite_psubset_induct
tff(fact_8149_even__set__encode__iff,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A3))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A3) ) ) ).

% even_set_encode_iff
tff(fact_8150_finite__has__maximal,axiom,
    ! [A3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ? [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A3)
            & ! [Xa: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A3)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8151_finite__has__maximal,axiom,
    ! [A3: set_real] :
      ( aa_set_real_o(finite_finite_real,A3)
     => ( ( A3 != bot_bot_set_real )
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),A3)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8152_finite__has__maximal,axiom,
    ! [A3: set_set_nat] :
      ( finite1152437895449049373et_nat(A3)
     => ( ( A3 != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( member_set_nat(X2,A3)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A3)
               => ( aa_set_nat_o(ord_less_eq_set_nat(X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8153_finite__has__maximal,axiom,
    ! [A3: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A3)
     => ( ( A3 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A3)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A3)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8154_finite__has__maximal,axiom,
    ! [A3: set_num] :
      ( aa_set_num_o(finite_finite_num,A3)
     => ( ( A3 != bot_bot_set_num )
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),A3)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A3)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8155_finite__has__maximal,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( A3 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A3)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8156_finite__has__maximal,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( A3 != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A3)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A3)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Xa)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8157_finite__has__minimal,axiom,
    ! [A3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ? [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A3)
            & ! [Xa: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A3)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8158_finite__has__minimal,axiom,
    ! [A3: set_real] :
      ( aa_set_real_o(finite_finite_real,A3)
     => ( ( A3 != bot_bot_set_real )
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),A3)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8159_finite__has__minimal,axiom,
    ! [A3: set_set_nat] :
      ( finite1152437895449049373et_nat(A3)
     => ( ( A3 != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( member_set_nat(X2,A3)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A3)
               => ( aa_set_nat_o(ord_less_eq_set_nat(Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8160_finite__has__minimal,axiom,
    ! [A3: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A3)
     => ( ( A3 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),A3)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A3)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8161_finite__has__minimal,axiom,
    ! [A3: set_num] :
      ( aa_set_num_o(finite_finite_num,A3)
     => ( ( A3 != bot_bot_set_num )
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),A3)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A3)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8162_finite__has__minimal,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( A3 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A3)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8163_finite__has__minimal,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( A3 != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),A3)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A3)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X2)
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8164_diff__preserves__multiset,axiom,
    ! [M8: fun_Pr5160692531401422620nt_nat,N2: fun_Pr5160692531401422620nt_nat] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_adm(M8)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun6302685600795359042_int_o(aTP_Lamm_adn(M8),N2))) ) ).

% diff_preserves_multiset
tff(fact_8165_diff__preserves__multiset,axiom,
    ! [M8: fun_nat_nat,N2: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_ado(M8)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_adp(M8),N2))) ) ).

% diff_preserves_multiset
tff(fact_8166_diff__preserves__multiset,axiom,
    ! [M8: fun_int_nat,N2: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_adq(M8)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_adr(M8),N2))) ) ).

% diff_preserves_multiset
tff(fact_8167_diff__preserves__multiset,axiom,
    ! [M8: fun_complex_nat,N2: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_ads(M8)))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun8723867720292580170plex_o(aTP_Lamm_adt(M8),N2))) ) ).

% diff_preserves_multiset
tff(fact_8168_diff__preserves__multiset,axiom,
    ! [M8: fun_Code_integer_nat,N2: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_adu(M8)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun4054834855839773292eger_o(aTP_Lamm_adv(M8),N2))) ) ).

% diff_preserves_multiset
tff(fact_8169_add__mset__in__multiset,axiom,
    ! [M8: fun_Pr5160692531401422620nt_nat,A2: product_prod_int_int] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_adm(M8)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_Pro3952998155787061599_int_o(aTP_Lamm_adw(M8),A2))) ) ).

% add_mset_in_multiset
tff(fact_8170_add__mset__in__multiset,axiom,
    ! [M8: fun_nat_nat,A2: nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_ado(M8)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_adx(M8),A2))) ) ).

% add_mset_in_multiset
tff(fact_8171_add__mset__in__multiset,axiom,
    ! [M8: fun_int_nat,A2: int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_adq(M8)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ady(M8),A2))) ) ).

% add_mset_in_multiset
tff(fact_8172_add__mset__in__multiset,axiom,
    ! [M8: fun_complex_nat,A2: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_ads(M8)))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_adz(M8),A2))) ) ).

% add_mset_in_multiset
tff(fact_8173_add__mset__in__multiset,axiom,
    ! [M8: fun_Code_integer_nat,A2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_adu(M8)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_aea(M8),A2))) ) ).

% add_mset_in_multiset
tff(fact_8174_finite__linorder__min__induct,axiom,
    ! [A3: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [B: code_integer,A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( ! [X3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A7)
                   => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),X3) )
               => ( aa_set5643387332196575111eger_o(P,A7)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(B,A7)) ) ) )
         => aa_set5643387332196575111eger_o(P,A3) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8175_finite__linorder__min__induct,axiom,
    ! [A3: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A3)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B: real,A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ! [X3: real] :
                    ( aa_set_real_o(member_real(X3),A7)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,B),X3) )
               => ( aa_set_real_o(P,A7)
                 => aa_set_real_o(P,insert_real(B,A7)) ) ) )
         => aa_set_real_o(P,A3) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8176_finite__linorder__min__induct,axiom,
    ! [A3: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A3)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B: rat,A7: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A7)
             => ( ! [X3: rat] :
                    ( aa_set_rat_o(member_rat(X3),A7)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),X3) )
               => ( aa_set_rat_o(P,A7)
                 => aa_set_rat_o(P,insert_rat(B,A7)) ) ) )
         => aa_set_rat_o(P,A3) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8177_finite__linorder__min__induct,axiom,
    ! [A3: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A3)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B: num,A7: set_num] :
              ( aa_set_num_o(finite_finite_num,A7)
             => ( ! [X3: num] :
                    ( aa_set_num_o(member_num(X3),A7)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,B),X3) )
               => ( aa_set_num_o(P,A7)
                 => aa_set_num_o(P,insert_num(B,A7)) ) ) )
         => aa_set_num_o(P,A3) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8178_finite__linorder__min__induct,axiom,
    ! [A3: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ! [X3: nat] :
                    ( aa_set_nat_o(member_nat(X3),A7)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),X3) )
               => ( aa_set_nat_o(P,A7)
                 => aa_set_nat_o(P,insert_nat(B,A7)) ) ) )
         => aa_set_nat_o(P,A3) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8179_finite__linorder__min__induct,axiom,
    ! [A3: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ! [X3: int] :
                    ( aa_set_int_o(member_int(X3),A7)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,B),X3) )
               => ( aa_set_int_o(P,A7)
                 => aa_set_int_o(P,insert_int(B,A7)) ) ) )
         => aa_set_int_o(P,A3) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8180_finite__linorder__max__induct,axiom,
    ! [A3: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [B: code_integer,A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( ! [X3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A7)
                   => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),B) )
               => ( aa_set5643387332196575111eger_o(P,A7)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(B,A7)) ) ) )
         => aa_set5643387332196575111eger_o(P,A3) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8181_finite__linorder__max__induct,axiom,
    ! [A3: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A3)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B: real,A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ! [X3: real] :
                    ( aa_set_real_o(member_real(X3),A7)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B) )
               => ( aa_set_real_o(P,A7)
                 => aa_set_real_o(P,insert_real(B,A7)) ) ) )
         => aa_set_real_o(P,A3) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8182_finite__linorder__max__induct,axiom,
    ! [A3: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A3)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B: rat,A7: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A7)
             => ( ! [X3: rat] :
                    ( aa_set_rat_o(member_rat(X3),A7)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),B) )
               => ( aa_set_rat_o(P,A7)
                 => aa_set_rat_o(P,insert_rat(B,A7)) ) ) )
         => aa_set_rat_o(P,A3) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8183_finite__linorder__max__induct,axiom,
    ! [A3: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A3)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B: num,A7: set_num] :
              ( aa_set_num_o(finite_finite_num,A7)
             => ( ! [X3: num] :
                    ( aa_set_num_o(member_num(X3),A7)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,X3),B) )
               => ( aa_set_num_o(P,A7)
                 => aa_set_num_o(P,insert_num(B,A7)) ) ) )
         => aa_set_num_o(P,A3) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8184_finite__linorder__max__induct,axiom,
    ! [A3: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ! [X3: nat] :
                    ( aa_set_nat_o(member_nat(X3),A7)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),B) )
               => ( aa_set_nat_o(P,A7)
                 => aa_set_nat_o(P,insert_nat(B,A7)) ) ) )
         => aa_set_nat_o(P,A3) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8185_finite__linorder__max__induct,axiom,
    ! [A3: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ! [X3: int] :
                    ( aa_set_int_o(member_int(X3),A7)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,X3),B) )
               => ( aa_set_int_o(P,A7)
                 => aa_set_int_o(P,insert_int(B,A7)) ) ) )
         => aa_set_int_o(P,A3) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8186_finite__ranking__induct,axiom,
    ! [S3: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o,Fa: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [X2: vEBT_VEBT,S5: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S5)
             => ( ! [Y4: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(Fa,Y4)),aa_VEBT_VEBT_rat(Fa,X2)) )
               => ( aa_set_VEBT_VEBT_o(P,S5)
                 => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X2,S5)) ) ) )
         => aa_set_VEBT_VEBT_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_8187_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: fun_set_complex_o,Fa: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X2: complex,S5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S5)
             => ( ! [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(Fa,Y4)),aa_complex_rat(Fa,X2)) )
               => ( aa_set_complex_o(P,S5)
                 => aa_set_complex_o(P,insert_complex(X2,S5)) ) ) )
         => aa_set_complex_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_8188_finite__ranking__induct,axiom,
    ! [S3: set_Code_integer,P: fun_se7289222521248229150eger_o,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [X2: code_integer,S5: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S5)
             => ( ! [Y4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(Fa,Y4)),aa_Code_integer_rat(Fa,X2)) )
               => ( aa_set5643387332196575111eger_o(P,S5)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(X2,S5)) ) ) )
         => aa_set5643387332196575111eger_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_8189_finite__ranking__induct,axiom,
    ! [S3: set_real,P: fun_set_real_o,Fa: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X2: real,S5: set_real] :
              ( aa_set_real_o(finite_finite_real,S5)
             => ( ! [Y4: real] :
                    ( aa_set_real_o(member_real(Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(Fa,Y4)),aa_real_rat(Fa,X2)) )
               => ( aa_set_real_o(P,S5)
                 => aa_set_real_o(P,insert_real(X2,S5)) ) ) )
         => aa_set_real_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_8190_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: fun_set_nat_o,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X2: nat,S5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S5)
             => ( ! [Y4: nat] :
                    ( aa_set_nat_o(member_nat(Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,Y4)),aa_nat_rat(Fa,X2)) )
               => ( aa_set_nat_o(P,S5)
                 => aa_set_nat_o(P,insert_nat(X2,S5)) ) ) )
         => aa_set_nat_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_8191_finite__ranking__induct,axiom,
    ! [S3: set_int,P: fun_set_int_o,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X2: int,S5: set_int] :
              ( aa_set_int_o(finite_finite_int,S5)
             => ( ! [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,Y4)),aa_int_rat(Fa,X2)) )
               => ( aa_set_int_o(P,S5)
                 => aa_set_int_o(P,insert_int(X2,S5)) ) ) )
         => aa_set_int_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_8192_finite__ranking__induct,axiom,
    ! [S3: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o,Fa: fun_VEBT_VEBT_num] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [X2: vEBT_VEBT,S5: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S5)
             => ( ! [Y4: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y4),S5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_VEBT_VEBT_num(Fa,Y4)),aa_VEBT_VEBT_num(Fa,X2)) )
               => ( aa_set_VEBT_VEBT_o(P,S5)
                 => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X2,S5)) ) ) )
         => aa_set_VEBT_VEBT_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_8193_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: fun_set_complex_o,Fa: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X2: complex,S5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S5)
             => ( ! [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),S5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_complex_num(Fa,Y4)),aa_complex_num(Fa,X2)) )
               => ( aa_set_complex_o(P,S5)
                 => aa_set_complex_o(P,insert_complex(X2,S5)) ) ) )
         => aa_set_complex_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_8194_finite__ranking__induct,axiom,
    ! [S3: set_Code_integer,P: fun_se7289222521248229150eger_o,Fa: fun_Code_integer_num] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [X2: code_integer,S5: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S5)
             => ( ! [Y4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Y4),S5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_Code_integer_num(Fa,Y4)),aa_Code_integer_num(Fa,X2)) )
               => ( aa_set5643387332196575111eger_o(P,S5)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(X2,S5)) ) ) )
         => aa_set5643387332196575111eger_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_8195_finite__ranking__induct,axiom,
    ! [S3: set_real,P: fun_set_real_o,Fa: fun_real_num] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X2: real,S5: set_real] :
              ( aa_set_real_o(finite_finite_real,S5)
             => ( ! [Y4: real] :
                    ( aa_set_real_o(member_real(Y4),S5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_real_num(Fa,Y4)),aa_real_num(Fa,X2)) )
               => ( aa_set_real_o(P,S5)
                 => aa_set_real_o(P,insert_real(X2,S5)) ) ) )
         => aa_set_real_o(P,S3) ) ) ) ).

% finite_ranking_induct
tff(fact_8196_ex__min__if__finite,axiom,
    ! [S3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( ( S3 != bot_bo3990330152332043303nteger )
       => ? [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S3)
            & ~ ? [Xa: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),S3)
                  & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8197_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( ( S3 != bot_bot_set_real )
       => ? [X2: real] :
            ( aa_set_real_o(member_real(X2),S3)
            & ~ ? [Xa: real] :
                  ( aa_set_real_o(member_real(Xa),S3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8198_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,S3)
     => ( ( S3 != bot_bot_set_rat )
       => ? [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),S3)
            & ~ ? [Xa: rat] :
                  ( aa_set_rat_o(member_rat(Xa),S3)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8199_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( aa_set_num_o(finite_finite_num,S3)
     => ( ( S3 != bot_bot_set_num )
       => ? [X2: num] :
            ( aa_set_num_o(member_num(X2),S3)
            & ~ ? [Xa: num] :
                  ( aa_set_num_o(member_num(Xa),S3)
                  & aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8200_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ( S3 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),S3)
            & ~ ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),S3)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8201_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ( S3 != bot_bot_set_int )
       => ? [X2: int] :
            ( aa_set_int_o(member_int(X2),S3)
            & ~ ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),S3)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X2) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8202_infinite__growing,axiom,
    ! [X5: set_Code_integer] :
      ( ( X5 != bot_bo3990330152332043303nteger )
     => ( ! [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),X5)
           => ? [Xa: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),X5)
                & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),Xa) ) )
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,X5) ) ) ).

% infinite_growing
tff(fact_8203_infinite__growing,axiom,
    ! [X5: set_real] :
      ( ( X5 != bot_bot_set_real )
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),X5)
           => ? [Xa: real] :
                ( aa_set_real_o(member_real(Xa),X5)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Xa) ) )
       => ~ aa_set_real_o(finite_finite_real,X5) ) ) ).

% infinite_growing
tff(fact_8204_infinite__growing,axiom,
    ! [X5: set_rat] :
      ( ( X5 != bot_bot_set_rat )
     => ( ! [X2: rat] :
            ( aa_set_rat_o(member_rat(X2),X5)
           => ? [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),X5)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Xa) ) )
       => ~ aa_set_rat_o(finite_finite_rat,X5) ) ) ).

% infinite_growing
tff(fact_8205_infinite__growing,axiom,
    ! [X5: set_num] :
      ( ( X5 != bot_bot_set_num )
     => ( ! [X2: num] :
            ( aa_set_num_o(member_num(X2),X5)
           => ? [Xa: num] :
                ( aa_set_num_o(member_num(Xa),X5)
                & aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Xa) ) )
       => ~ aa_set_num_o(finite_finite_num,X5) ) ) ).

% infinite_growing
tff(fact_8206_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),X5)
           => ? [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),X5)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Xa) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X5) ) ) ).

% infinite_growing
tff(fact_8207_infinite__growing,axiom,
    ! [X5: set_int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),X5)
           => ? [Xa: int] :
                ( aa_set_int_o(member_int(Xa),X5)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Xa) ) )
       => ~ aa_set_int_o(finite_finite_int,X5) ) ) ).

% infinite_growing
tff(fact_8208_filter__preserves__multiset,axiom,
    ! [M8: fun_Pr5160692531401422620nt_nat,P: fun_Pr3238374608701176162_int_o] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_adm(M8)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_aeb(M8),P))) ) ).

% filter_preserves_multiset
tff(fact_8209_filter__preserves__multiset,axiom,
    ! [M8: fun_nat_nat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_ado(M8)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aec(M8),P))) ) ).

% filter_preserves_multiset
tff(fact_8210_filter__preserves__multiset,axiom,
    ! [M8: fun_int_nat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_adq(M8)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_aed(M8),P))) ) ).

% filter_preserves_multiset
tff(fact_8211_filter__preserves__multiset,axiom,
    ! [M8: fun_complex_nat,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_ads(M8)))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_aee(M8),P))) ) ).

% filter_preserves_multiset
tff(fact_8212_filter__preserves__multiset,axiom,
    ! [M8: fun_Code_integer_nat,P: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_adu(M8)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_aef(M8),P))) ) ).

% filter_preserves_multiset
tff(fact_8213_infinite__int__iff__unbounded,axiom,
    ! [S3: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S3)
    <=> ! [M6: int] :
        ? [N8: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,M6),aa_int_int(abs_abs_int,N8))
          & aa_set_int_o(member_int(N8),S3) ) ) ).

% infinite_int_iff_unbounded
tff(fact_8214_infinite__int__iff__unbounded__le,axiom,
    ! [S3: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S3)
    <=> ! [M6: int] :
        ? [N8: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M6),aa_int_int(abs_abs_int,N8))
          & aa_set_int_o(member_int(N8),S3) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_8215_cos__arcsin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xb)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_8216_arcsin__0,axiom,
    aa_real_real(arcsin,zero_zero_real) = zero_zero_real ).

% arcsin_0
tff(fact_8217_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% arcsin_1
tff(fact_8218_arcsin__minus__1,axiom,
    aa_real_real(arcsin,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ).

% arcsin_minus_1
tff(fact_8219_arcsin__less__arcsin,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_8220_arcsin__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_8221_cos__arcsin__nonzero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xb)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_8222_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
    <=> ! [M6: nat] :
        ? [N8: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M6),N8)
          & aa_set_nat_o(member_nat(N8),S3) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_8223_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M3)
         => ? [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N4)
              & aa_set_nat_o(member_nat(N4),S3) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S3) ) ).

% unbounded_k_infinite
tff(fact_8224_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
    <=> ! [M6: nat] :
        ? [N8: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M6),N8)
          & aa_set_nat_o(member_nat(N8),S3) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_8225_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_8226_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_8227_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ).

% arcsin_ubound
tff(fact_8228_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_bounded
tff(fact_8229_arcsin__sin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,Xb)) = Xb ) ) ) ).

% arcsin_sin
tff(fact_8230_le__arcsin__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(arcsin,Xb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),Xb) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_8231_arcsin__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),Y)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sin_real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_8232_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),pi)
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_8233_arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_8234_vebt__buildup_Oelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half))),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half))),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_8235_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,Xb: fun_VEBT_VEBT_uint32,Y: fun_VEBT_VEBT_uint32] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun6272599145770227614VEBT_o(aTP_Lamm_aeg(I5),Xb)))
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun6272599145770227614VEBT_o(aTP_Lamm_aeg(I5),Y)))
       => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun6272599145770227614VEBT_o(aa_fun1156574460560677430VEBT_o(aTP_Lamm_aeh(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8236_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xb: fun_real_uint32,Y: fun_real_uint32] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun7556296611544133298real_o(aTP_Lamm_aei(I5),Xb)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun7556296611544133298real_o(aTP_Lamm_aei(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun7556296611544133298real_o(aa_fun5572330763077102134real_o(aTP_Lamm_aej(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8237_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xb: fun_nat_uint32,Y: fun_nat_uint32] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1182053767699555178_nat_o(aTP_Lamm_aek(I5),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1182053767699555178_nat_o(aTP_Lamm_aek(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1182053767699555178_nat_o(aa_fun6957237400125433034_nat_o(aTP_Lamm_ael(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8238_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xb: fun_int_uint32,Y: fun_int_uint32] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun6992502331602369202_int_o(aTP_Lamm_aem(I5),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun6992502331602369202_int_o(aTP_Lamm_aem(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun6992502331602369202_int_o(aa_fun3369613587901038774_int_o(aTP_Lamm_aen(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8239_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,Xb: fun_complex_uint32,Y: fun_complex_uint32] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun436548534304416686plex_o(aTP_Lamm_aeo(I5),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun436548534304416686plex_o(aTP_Lamm_aeo(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun436548534304416686plex_o(aa_fun2751405167760845232plex_o(aTP_Lamm_aep(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8240_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,Xb: fun_Co1554742298434871494uint32,Y: fun_Co1554742298434871494uint32] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8926496376162642256eger_o(aTP_Lamm_aeq(I5),Xb)))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8926496376162642256eger_o(aTP_Lamm_aeq(I5),Y)))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8926496376162642256eger_o(aa_fun1119956648780089443eger_o(aTP_Lamm_aer(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8241_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,Xb: fun_VEBT_VEBT_real,Y: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aTP_Lamm_aes(I5),Xb)))
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aTP_Lamm_aes(I5),Y)))
       => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aa_fun5637434397837810486VEBT_o(aTP_Lamm_aet(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8242_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xb: fun_real_real,Y: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_aeu(I5),Xb)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_aeu(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_aev(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8243_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xb: fun_nat_real,Y: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_aew(I5),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_aew(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_aex(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8244_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xb: fun_int_real,Y: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_aey(I5),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_aey(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_aez(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_8245_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,Xb: fun_VEBT_VEBT_real,Y: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aTP_Lamm_afa(I5),Xb)))
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aTP_Lamm_afa(I5),Y)))
       => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun1965331558590768990VEBT_o(aa_fun5637434397837810486VEBT_o(aTP_Lamm_afb(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8246_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xb: fun_real_real,Y: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_afc(I5),Xb)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_afc(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_afd(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8247_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xb: fun_nat_real,Y: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_afe(I5),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_afe(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_aff(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8248_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xb: fun_int_real,Y: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_afg(I5),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_afg(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_afh(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8249_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,Xb: fun_complex_real,Y: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_afi(I5),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_afi(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_afj(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8250_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,Xb: fun_Co3960896333048504454r_real,Y: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun3640226575523069072eger_o(aTP_Lamm_afk(I5),Xb)))
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun3640226575523069072eger_o(aTP_Lamm_afk(I5),Y)))
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun3640226575523069072eger_o(aa_fun5110011637356352227eger_o(aTP_Lamm_afl(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8251_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,Xb: fun_VEBT_VEBT_rat,Y: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5889122065779655690VEBT_o(aTP_Lamm_afm(I5),Xb)))
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5889122065779655690VEBT_o(aTP_Lamm_afm(I5),Y)))
       => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,collect_VEBT_VEBT(aa_fun5889122065779655690VEBT_o(aa_fun1978406156034293430VEBT_o(aTP_Lamm_afn(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8252_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xb: fun_real_rat,Y: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_afo(I5),Xb)))
     => ( aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_afo(I5),Y)))
       => aa_set_real_o(finite_finite_real,collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_afp(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8253_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xb: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_afq(I5),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_afq(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_afr(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8254_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xb: fun_int_rat,Y: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_afs(I5),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_afs(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_aft(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_8255_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(B2),Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(B2),aa_nat_real(semiri5074537144036343181t_real,K))),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(powr_real(B2),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_8256_intind,axiom,
    ! [I: nat,N: nat,P: fun_nat_o,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),N)
     => ( aa_nat_o(P,Xb)
       => aa_nat_o(P,aa_nat_nat(nth_nat(replicate_nat(N,Xb)),I)) ) ) ).

% intind
tff(fact_8257_intind,axiom,
    ! [I: nat,N: nat,P: fun_VEBT_VEBTi_o,Xb: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),N)
     => ( aa_VEBT_VEBTi_o(P,Xb)
       => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(replicate_VEBT_VEBTi(N,Xb),I)) ) ) ).

% intind
tff(fact_8258_intind,axiom,
    ! [I: nat,N: nat,P: fun_int_o,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),N)
     => ( aa_int_o(P,Xb)
       => aa_int_o(P,aa_nat_int(nth_int(replicate_int(N,Xb)),I)) ) ) ).

% intind
tff(fact_8259_intind,axiom,
    ! [I: nat,N: nat,P: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),N)
     => ( aa_VEBT_VEBT_o(P,Xb)
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(N,Xb),I)) ) ) ).

% intind
tff(fact_8260_powr__0,axiom,
    ! [Z: real] : aa_real_real(powr_real(zero_zero_real),Z) = zero_zero_real ).

% powr_0
tff(fact_8261_powr__eq__0__iff,axiom,
    ! [W: real,Z: real] :
      ( ( aa_real_real(powr_real(W),Z) = zero_zero_real )
    <=> ( W = zero_zero_real ) ) ).

% powr_eq_0_iff
tff(fact_8262_powr__one__eq__one,axiom,
    ! [A2: real] : aa_real_real(powr_real(one_one_real),A2) = one_one_real ).

% powr_one_eq_one
tff(fact_8263_replicate__eq__replicate,axiom,
    ! [M: nat,Xb: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(M,Xb) = replicate_VEBT_VEBT(N,Y) )
    <=> ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( Xb = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_8264_length__replicate,axiom,
    ! [N: nat,Xb: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(N,Xb)) = N ).

% length_replicate
tff(fact_8265_length__replicate,axiom,
    ! [N: nat,Xb: real] : size_size_list_real(replicate_real(N,Xb)) = N ).

% length_replicate
tff(fact_8266_length__replicate,axiom,
    ! [N: nat,Xb: $o] : size_size_list_o(replicate_o(N,(Xb))) = N ).

% length_replicate
tff(fact_8267_length__replicate,axiom,
    ! [N: nat,Xb: nat] : size_size_list_nat(replicate_nat(N,Xb)) = N ).

% length_replicate
tff(fact_8268_length__replicate,axiom,
    ! [N: nat,Xb: int] : size_size_list_int(replicate_int(N,Xb)) = N ).

% length_replicate
tff(fact_8269_powr__zero__eq__one,axiom,
    ! [Xb: real] :
      aa_real_real(powr_real(Xb),zero_zero_real) = $ite(Xb = zero_zero_real,zero_zero_real,one_one_real) ).

% powr_zero_eq_one
tff(fact_8270_in__set__replicate,axiom,
    ! [Xb: nat,N: nat,Y: nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(replicate_nat(N,Y)))
    <=> ( ( Xb = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8271_in__set__replicate,axiom,
    ! [Xb: real,N: nat,Y: real] :
      ( aa_set_real_o(member_real(Xb),set_real2(replicate_real(N,Y)))
    <=> ( ( Xb = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8272_in__set__replicate,axiom,
    ! [Xb: int,N: nat,Y: int] :
      ( aa_set_int_o(member_int(Xb),set_int2(replicate_int(N,Y)))
    <=> ( ( Xb = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8273_in__set__replicate,axiom,
    ! [Xb: produc6575502325842934193n_assn,N: nat,Y: produc6575502325842934193n_assn] :
      ( member7957490590177025114n_assn(Xb,set_Pr1139785259514867910n_assn(replic3825545231534752113n_assn(N,Y)))
    <=> ( ( Xb = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8274_in__set__replicate,axiom,
    ! [Xb: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xb),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,Y)))
    <=> ( ( Xb = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_8275_Bex__set__replicate,axiom,
    ! [N: nat,A2: nat,P: fun_nat_o] :
      ( ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(replicate_nat(N,A2)))
          & aa_nat_o(P,X) )
    <=> ( aa_nat_o(P,A2)
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_8276_Bex__set__replicate,axiom,
    ! [N: nat,A2: real,P: fun_real_o] :
      ( ? [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(replicate_real(N,A2)))
          & aa_real_o(P,X) )
    <=> ( aa_real_o(P,A2)
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_8277_Bex__set__replicate,axiom,
    ! [N: nat,A2: int,P: fun_int_o] :
      ( ? [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(replicate_int(N,A2)))
          & aa_int_o(P,X) )
    <=> ( aa_int_o(P,A2)
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_8278_Bex__set__replicate,axiom,
    ! [N: nat,A2: produc6575502325842934193n_assn,P: fun_Pr3571187171380813162assn_o] :
      ( ? [X: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(replic3825545231534752113n_assn(N,A2)))
          & aa_Pro2491721552408928739assn_o(P,X) )
    <=> ( aa_Pro2491721552408928739assn_o(P,A2)
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_8279_Bex__set__replicate,axiom,
    ! [N: nat,A2: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ? [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A2)))
          & aa_VEBT_VEBT_o(P,X) )
    <=> ( aa_VEBT_VEBT_o(P,A2)
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_8280_Ball__set__replicate,axiom,
    ! [N: nat,A2: nat,P: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(replicate_nat(N,A2)))
         => aa_nat_o(P,X) )
    <=> ( aa_nat_o(P,A2)
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_8281_Ball__set__replicate,axiom,
    ! [N: nat,A2: real,P: fun_real_o] :
      ( ! [X: real] :
          ( aa_set_real_o(member_real(X),set_real2(replicate_real(N,A2)))
         => aa_real_o(P,X) )
    <=> ( aa_real_o(P,A2)
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_8282_Ball__set__replicate,axiom,
    ! [N: nat,A2: int,P: fun_int_o] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(replicate_int(N,A2)))
         => aa_int_o(P,X) )
    <=> ( aa_int_o(P,A2)
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_8283_Ball__set__replicate,axiom,
    ! [N: nat,A2: produc6575502325842934193n_assn,P: fun_Pr3571187171380813162assn_o] :
      ( ! [X: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(replic3825545231534752113n_assn(N,A2)))
         => aa_Pro2491721552408928739assn_o(P,X) )
    <=> ( aa_Pro2491721552408928739assn_o(P,A2)
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_8284_Ball__set__replicate,axiom,
    ! [N: nat,A2: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(replicate_VEBT_VEBT(N,A2)))
         => aa_VEBT_VEBT_o(P,X) )
    <=> ( aa_VEBT_VEBT_o(P,A2)
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_8285_powr__gt__zero,axiom,
    ! [Xb: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(powr_real(Xb),A2))
    <=> ( Xb != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_8286_powr__nonneg__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(A2),Xb)),zero_zero_real)
    <=> ( A2 = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_8287_nth__replicate,axiom,
    ! [I: nat,N: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),N)
     => ( aa_nat_nat(nth_nat(replicate_nat(N,Xb)),I) = Xb ) ) ).

% nth_replicate
tff(fact_8288_nth__replicate,axiom,
    ! [I: nat,N: nat,Xb: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),N)
     => ( nth_VEBT_VEBTi(replicate_VEBT_VEBTi(N,Xb),I) = Xb ) ) ).

% nth_replicate
tff(fact_8289_nth__replicate,axiom,
    ! [I: nat,N: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),N)
     => ( aa_nat_int(nth_int(replicate_int(N,Xb)),I) = Xb ) ) ).

% nth_replicate
tff(fact_8290_nth__replicate,axiom,
    ! [I: nat,N: nat,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),N)
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(N,Xb),I) = Xb ) ) ).

% nth_replicate
tff(fact_8291_powr__less__cancel__iff,axiom,
    ! [Xb: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xb),A2)),aa_real_real(powr_real(Xb),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% powr_less_cancel_iff
tff(fact_8292_powr__eq__one__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( ( aa_real_real(powr_real(A2),Xb) = one_one_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_8293_powr__one__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(powr_real(Xb),one_one_real) = Xb )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% powr_one_gt_zero_iff
tff(fact_8294_powr__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),one_one_real) = Xb ) ) ).

% powr_one
tff(fact_8295_powr__le__cancel__iff,axiom,
    ! [Xb: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xb),A2)),aa_real_real(powr_real(Xb),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ).

% powr_le_cancel_iff
tff(fact_8296_numeral__powr__numeral__real,axiom,
    ! [M: num,N: num] : aa_real_real(powr_real(numeral_numeral_real(M)),numeral_numeral_real(N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(M)),aa_num_nat(numeral_numeral_nat,N)) ).

% numeral_powr_numeral_real
tff(fact_8297_set__replicate,axiom,
    ! [N: nat,Xb: produc6575502325842934193n_assn] :
      ( ( N != zero_zero_nat )
     => ( set_Pr1139785259514867910n_assn(replic3825545231534752113n_assn(N,Xb)) = insert5290817439147925377n_assn(Xb,bot_bo1176836662018730877n_assn) ) ) ).

% set_replicate
tff(fact_8298_set__replicate,axiom,
    ! [N: nat,Xb: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(N,Xb)) = insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_8299_set__replicate,axiom,
    ! [N: nat,Xb: real] :
      ( ( N != zero_zero_nat )
     => ( set_real2(replicate_real(N,Xb)) = insert_real(Xb,bot_bot_set_real) ) ) ).

% set_replicate
tff(fact_8300_set__replicate,axiom,
    ! [N: nat,Xb: nat] :
      ( ( N != zero_zero_nat )
     => ( set_nat2(replicate_nat(N,Xb)) = insert_nat(Xb,bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_8301_set__replicate,axiom,
    ! [N: nat,Xb: int] :
      ( ( N != zero_zero_nat )
     => ( set_int2(replicate_int(N,Xb)) = insert_int(Xb,bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_8302_powr__log__cancel,axiom,
    ! [A2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(powr_real(A2),aa_real_real(log(A2),Xb)) = Xb ) ) ) ) ).

% powr_log_cancel
tff(fact_8303_log__powr__cancel,axiom,
    ! [A2: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_real(log(A2),aa_real_real(powr_real(A2),Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_8304_powr__numeral,axiom,
    ! [Xb: real,N: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),numeral_numeral_real(N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,N)) ) ) ).

% powr_numeral
tff(fact_8305_map__fst__mk__fst,axiom,
    ! [K: num,L: list_num] : map_Pr454908937103039467um_num(product_fst_num_num,map_nu2851882102140640437um_num(product_Pair_num_num(K),L)) = replicate_num(size_size_list_num(L),K) ).

% map_fst_mk_fst
tff(fact_8306_map__fst__mk__fst,axiom,
    ! [K: nat,L: list_P6730324909620535719T_VEBT] : map_Pr3018521781701129308BT_nat(produc758997459209783180T_VEBT,map_Pr7909041933469137955T_VEBT(produc1750349459881913976T_VEBT(K),L)) = replicate_nat(size_s4764337671732037139T_VEBT(L),K) ).

% map_fst_mk_fst
tff(fact_8307_map__fst__mk__fst,axiom,
    ! [K: nat,L: list_num] : map_Pr5956769322976601943um_nat(product_fst_nat_num,map_nu4721551698833171051at_num(product_Pair_nat_num(K),L)) = replicate_nat(size_size_list_num(L),K) ).

% map_fst_mk_fst
tff(fact_8308_map__fst__mk__fst,axiom,
    ! [K: vEBT_VEBT,L: list_real] : map_Pr6195879527588727455T_VEBT(produc8110914911036349469T_real,map_re8618229306769252225T_real(produc8117437818029410057T_real(K),L)) = replicate_VEBT_VEBT(size_size_list_real(L),K) ).

% map_fst_mk_fst
tff(fact_8309_map__fst__mk__fst,axiom,
    ! [K: vEBT_VEBT,L: list_o] : map_Pr4868735216952053677T_VEBT(produc4993121158135996263VEBT_o,map_o_6754667662019005495VEBT_o(produc8721562602347293563VEBT_o(K),L)) = replicate_VEBT_VEBT(size_size_list_o(L),K) ).

% map_fst_mk_fst
tff(fact_8310_map__fst__mk__fst,axiom,
    ! [K: vEBT_VEBT,L: list_nat] : map_Pr1380729192516676091T_VEBT(produc8713918199166443969BT_nat,map_na4631810538828370761BT_nat(produc738532404422230701BT_nat(K),L)) = replicate_VEBT_VEBT(size_size_list_nat(L),K) ).

% map_fst_mk_fst
tff(fact_8311_map__fst__mk__fst,axiom,
    ! [K: vEBT_VEBT,L: list_int] : map_Pr3257657825534036127T_VEBT(produc8711427728657393693BT_int,map_in8151279748432256513BT_int(produc736041933913180425BT_int(K),L)) = replicate_VEBT_VEBT(size_size_list_int(L),K) ).

% map_fst_mk_fst
tff(fact_8312_map__fst__mk__fst,axiom,
    ! [K: nat,L: list_nat] : map_Pr3938374229010428429at_nat(product_fst_nat_nat,map_na7298421622053143531at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,K),L)) = replicate_nat(size_size_list_nat(L),K) ).

% map_fst_mk_fst
tff(fact_8313_map__fst__mk__fst,axiom,
    ! [K: int,L: list_int] : map_Pr6494458893431244577nt_int(product_fst_int_int,map_in7157766398909135175nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K),L)) = replicate_int(size_size_list_int(L),K) ).

% map_fst_mk_fst
tff(fact_8314_map__fst__mk__fst,axiom,
    ! [K: assn,L: list_assn] : map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,map_as2373307505041272643n_assn(produc118845697133431529n_assn(K),L)) = replicate_assn(size_size_list_assn(L),K) ).

% map_fst_mk_fst
tff(fact_8315_map__snd__mk__snd,axiom,
    ! [K: num,L: list_num] : map_Pr454908937103039467um_num(product_snd_num_num,map_nu2851882102140640437um_num(aTP_Lamm_mr(K),L)) = replicate_num(size_size_list_num(L),K) ).

% map_snd_mk_snd
tff(fact_8316_map__snd__mk__snd,axiom,
    ! [K: vEBT_VEBT,L: list_real] : map_Pr6147841162850987569T_VEBT(produc5083336317046741121T_VEBT,map_re7205069664741861231T_VEBT(aTP_Lamm_afu(K),L)) = replicate_VEBT_VEBT(size_size_list_real(L),K) ).

% map_snd_mk_snd
tff(fact_8317_map__snd__mk__snd,axiom,
    ! [K: vEBT_VEBT,L: list_o] : map_Pr7652832201708611317T_VEBT(produc7938581201502569057T_VEBT,map_o_8925299737569714927T_VEBT(aTP_Lamm_afv(K),L)) = replicate_VEBT_VEBT(size_size_list_o(L),K) ).

% map_snd_mk_snd
tff(fact_8318_map__snd__mk__snd,axiom,
    ! [K: vEBT_VEBT,L: list_nat] : map_Pr8570210702748812117T_VEBT(produc8172668247895388509T_VEBT,map_na3584885621601055599T_VEBT(aTP_Lamm_afw(K),L)) = replicate_VEBT_VEBT(size_size_list_nat(L),K) ).

% map_snd_mk_snd
tff(fact_8319_map__snd__mk__snd,axiom,
    ! [K: produc4813437837504472865T_VEBT,L: list_nat] : map_Pr3651661567648760725T_VEBT(produc2084898568784432842T_VEBT,map_na3322839687800594908T_VEBT(aTP_Lamm_ms(K),L)) = replic862811827245231841T_VEBT(size_size_list_nat(L),K) ).

% map_snd_mk_snd
tff(fact_8320_map__snd__mk__snd,axiom,
    ! [K: num,L: list_nat] : map_Pr2514101109132380577um_num(product_snd_nat_num,map_na8006665559001981237at_num(aTP_Lamm_mt(K),L)) = replicate_num(size_size_list_nat(L),K) ).

% map_snd_mk_snd
tff(fact_8321_map__snd__mk__snd,axiom,
    ! [K: vEBT_VEBT,L: list_int] : map_Pr1314269154781486001T_VEBT(produc1678900780639429121T_VEBT,map_in4788438383458178671T_VEBT(aTP_Lamm_afx(K),L)) = replicate_VEBT_VEBT(size_size_list_int(L),K) ).

% map_snd_mk_snd
tff(fact_8322_map__snd__mk__snd,axiom,
    ! [K: nat,L: list_nat] : map_Pr3938374229010428429at_nat(product_snd_nat_nat,map_na7298421622053143531at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_mu,K),L)) = replicate_nat(size_size_list_nat(L),K) ).

% map_snd_mk_snd
tff(fact_8323_map__snd__mk__snd,axiom,
    ! [K: int,L: list_int] : map_Pr6494458893431244577nt_int(product_snd_int_int,map_in7157766398909135175nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_mv,K),L)) = replicate_int(size_size_list_int(L),K) ).

% map_snd_mk_snd
tff(fact_8324_map__snd__mk__snd,axiom,
    ! [K: assn,L: list_assn] : map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,map_as2373307505041272643n_assn(aTP_Lamm_mw(K),L)) = replicate_assn(size_size_list_assn(L),K) ).

% map_snd_mk_snd
tff(fact_8325_square__powr__half,axiom,
    ! [Xb: real] : aa_real_real(powr_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xb) ).

% square_powr_half
tff(fact_8326_powr__less__mono2__neg,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Y),A2)),aa_real_real(powr_real(Xb),A2)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_8327_powr__non__neg,axiom,
    ! [A2: real,Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(A2),Xb)),zero_zero_real) ).

% powr_non_neg
tff(fact_8328_powr__mono2,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xb),A2)),aa_real_real(powr_real(Y),A2)) ) ) ) ).

% powr_mono2
tff(fact_8329_powr__ge__pzero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(powr_real(Xb),Y)) ).

% powr_ge_pzero
tff(fact_8330_powr__less__cancel,axiom,
    ! [Xb: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xb),A2)),aa_real_real(powr_real(Xb),B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% powr_less_cancel
tff(fact_8331_powr__less__mono,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xb),A2)),aa_real_real(powr_real(Xb),B2)) ) ) ).

% powr_less_mono
tff(fact_8332_replicate__length__same,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Xb: produc6575502325842934193n_assn] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(Xs))
         => ( X2 = Xb ) )
     => ( replic3825545231534752113n_assn(size_s6829681357464350627n_assn(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_8333_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => ( X2 = Xb ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_8334_replicate__length__same,axiom,
    ! [Xs: list_real,Xb: real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => ( X2 = Xb ) )
     => ( replicate_real(size_size_list_real(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_8335_replicate__length__same,axiom,
    ! [Xs: list_o,Xb: $o] :
      ( ! [X2: $o] :
          ( member_o((X2),set_o2(Xs))
         => ( (X2)
          <=> (Xb) ) )
     => ( replicate_o(size_size_list_o(Xs),(Xb)) = Xs ) ) ).

% replicate_length_same
tff(fact_8336_replicate__length__same,axiom,
    ! [Xs: list_nat,Xb: nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => ( X2 = Xb ) )
     => ( replicate_nat(size_size_list_nat(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_8337_replicate__length__same,axiom,
    ! [Xs: list_int,Xb: int] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),set_int2(Xs))
         => ( X2 = Xb ) )
     => ( replicate_int(size_size_list_int(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_8338_replicate__eqI,axiom,
    ! [Xs: list_P8527749157015355191n_assn,N: nat,Xb: produc6575502325842934193n_assn] :
      ( ( size_s6829681357464350627n_assn(Xs) = N )
     => ( ! [Y2: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(Y2,set_Pr1139785259514867910n_assn(Xs))
           => ( Y2 = Xb ) )
       => ( Xs = replic3825545231534752113n_assn(N,Xb) ) ) ) ).

% replicate_eqI
tff(fact_8339_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat,Xb: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = N )
     => ( ! [Y2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y2),set_VEBT_VEBT2(Xs))
           => ( Y2 = Xb ) )
       => ( Xs = replicate_VEBT_VEBT(N,Xb) ) ) ) ).

% replicate_eqI
tff(fact_8340_replicate__eqI,axiom,
    ! [Xs: list_real,N: nat,Xb: real] :
      ( ( size_size_list_real(Xs) = N )
     => ( ! [Y2: real] :
            ( aa_set_real_o(member_real(Y2),set_real2(Xs))
           => ( Y2 = Xb ) )
       => ( Xs = replicate_real(N,Xb) ) ) ) ).

% replicate_eqI
tff(fact_8341_replicate__eqI,axiom,
    ! [Xs: list_o,N: nat,Xb: $o] :
      ( ( size_size_list_o(Xs) = N )
     => ( ! [Y2: $o] :
            ( member_o((Y2),set_o2(Xs))
           => ( (Y2)
            <=> (Xb) ) )
       => ( Xs = replicate_o(N,(Xb)) ) ) ) ).

% replicate_eqI
tff(fact_8342_replicate__eqI,axiom,
    ! [Xs: list_nat,N: nat,Xb: nat] :
      ( ( size_size_list_nat(Xs) = N )
     => ( ! [Y2: nat] :
            ( aa_set_nat_o(member_nat(Y2),set_nat2(Xs))
           => ( Y2 = Xb ) )
       => ( Xs = replicate_nat(N,Xb) ) ) ) ).

% replicate_eqI
tff(fact_8343_replicate__eqI,axiom,
    ! [Xs: list_int,N: nat,Xb: int] :
      ( ( size_size_list_int(Xs) = N )
     => ( ! [Y2: int] :
            ( aa_set_int_o(member_int(Y2),set_int2(Xs))
           => ( Y2 = Xb ) )
       => ( Xs = replicate_int(N,Xb) ) ) ) ).

% replicate_eqI
tff(fact_8344_map__replicate__const,axiom,
    ! [K: nat,Lst: list_VEBT_VEBT] : map_VEBT_VEBT_nat(aTP_Lamm_afy(K),Lst) = replicate_nat(size_s6755466524823107622T_VEBT(Lst),K) ).

% map_replicate_const
tff(fact_8345_map__replicate__const,axiom,
    ! [K: real,Lst: list_VEBT_VEBT] : map_VEBT_VEBT_real(aTP_Lamm_afz(K),Lst) = replicate_real(size_s6755466524823107622T_VEBT(Lst),K) ).

% map_replicate_const
tff(fact_8346_map__replicate__const,axiom,
    ! [K: assn,Lst: list_P8527749157015355191n_assn] : map_Pr8991440229025900053n_assn(aTP_Lamm_aga(K),Lst) = replicate_assn(size_s6829681357464350627n_assn(Lst),K) ).

% map_replicate_const
tff(fact_8347_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_real] : map_real_VEBT_VEBT(aTP_Lamm_agb(K),Lst) = replicate_VEBT_VEBT(size_size_list_real(Lst),K) ).

% map_replicate_const
tff(fact_8348_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_o] : map_o_VEBT_VEBT(aTP_Lamm_agc(K),Lst) = replicate_VEBT_VEBT(size_size_list_o(Lst),K) ).

% map_replicate_const
tff(fact_8349_map__replicate__const,axiom,
    ! [K: nat,Lst: list_nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dq,K),Lst) = replicate_nat(size_size_list_nat(Lst),K) ).

% map_replicate_const
tff(fact_8350_map__replicate__const,axiom,
    ! [K: $o,Lst: list_nat] : map_nat_o(aa_o_fun_nat_o(aTP_Lamm_agd,(K)),Lst) = replicate_o(size_size_list_nat(Lst),(K)) ).

% map_replicate_const
tff(fact_8351_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_nat] : map_nat_VEBT_VEBT(aTP_Lamm_age(K),Lst) = replicate_VEBT_VEBT(size_size_list_nat(Lst),K) ).

% map_replicate_const
tff(fact_8352_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_int] : map_int_VEBT_VEBT(aTP_Lamm_agf(K),Lst) = replicate_VEBT_VEBT(size_size_list_int(Lst),K) ).

% map_replicate_const
tff(fact_8353_powr__less__mono2,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xb),A2)),aa_real_real(powr_real(Y),A2)) ) ) ) ).

% powr_less_mono2
tff(fact_8354_powr__mono2_H,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Y),A2)),aa_real_real(powr_real(Xb),A2)) ) ) ) ).

% powr_mono2'
tff(fact_8355_gr__one__powr,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(powr_real(Xb),Y)) ) ) ).

% gr_one_powr
tff(fact_8356_powr__inj,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( ( aa_real_real(powr_real(A2),Xb) = aa_real_real(powr_real(A2),Y) )
        <=> ( Xb = Y ) ) ) ) ).

% powr_inj
tff(fact_8357_ge__one__powr__ge__zero,axiom,
    ! [Xb: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(powr_real(Xb),A2)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_8358_powr__mono__both,axiom,
    ! [A2: real,B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xb),A2)),aa_real_real(powr_real(Y),B2)) ) ) ) ) ).

% powr_mono_both
tff(fact_8359_powr__le1,axiom,
    ! [A2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xb),A2)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_8360_powr__divide,axiom,
    ! [Xb: real,Y: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(powr_real(aa_real_real(divide_divide_real(Xb),Y)),A2) = aa_real_real(divide_divide_real(aa_real_real(powr_real(Xb),A2)),aa_real_real(powr_real(Y),A2)) ) ) ) ).

% powr_divide
tff(fact_8361_powr__mult,axiom,
    ! [Xb: real,Y: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(powr_real(aa_real_real(times_times_real(Xb),Y)),A2) = aa_real_real(times_times_real(aa_real_real(powr_real(Xb),A2)),aa_real_real(powr_real(Y),A2)) ) ) ) ).

% powr_mult
tff(fact_8362_log__base__powr,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(log(aa_real_real(powr_real(A2),B2)),Xb) = aa_real_real(divide_divide_real(aa_real_real(log(A2),Xb)),B2) ) ) ).

% log_base_powr
tff(fact_8363_log__powr,axiom,
    ! [Xb: real,B2: real,Y: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(log(B2),aa_real_real(powr_real(Xb),Y)) = aa_real_real(times_times_real(Y),aa_real_real(log(B2),Xb)) ) ) ).

% log_powr
tff(fact_8364_ln__powr,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(powr_real(Xb),Y)) = aa_real_real(times_times_real(Y),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_powr
tff(fact_8365_powr__diff,axiom,
    ! [W: real,Z1: real,Z22: real] : aa_real_real(powr_real(W),aa_real_real(minus_minus_real(Z1),Z22)) = aa_real_real(divide_divide_real(aa_real_real(powr_real(W),Z1)),aa_real_real(powr_real(W),Z22)) ).

% powr_diff
tff(fact_8366_set__replicate__conv__if,axiom,
    ! [N: nat,Xb: produc6575502325842934193n_assn] :
      set_Pr1139785259514867910n_assn(replic3825545231534752113n_assn(N,Xb)) = $ite(N = zero_zero_nat,bot_bo1176836662018730877n_assn,insert5290817439147925377n_assn(Xb,bot_bo1176836662018730877n_assn)) ).

% set_replicate_conv_if
tff(fact_8367_set__replicate__conv__if,axiom,
    ! [N: nat,Xb: vEBT_VEBT] :
      set_VEBT_VEBT2(replicate_VEBT_VEBT(N,Xb)) = $ite(N = zero_zero_nat,bot_bo8194388402131092736T_VEBT,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)) ).

% set_replicate_conv_if
tff(fact_8368_set__replicate__conv__if,axiom,
    ! [N: nat,Xb: real] :
      set_real2(replicate_real(N,Xb)) = $ite(N = zero_zero_nat,bot_bot_set_real,insert_real(Xb,bot_bot_set_real)) ).

% set_replicate_conv_if
tff(fact_8369_set__replicate__conv__if,axiom,
    ! [N: nat,Xb: nat] :
      set_nat2(replicate_nat(N,Xb)) = $ite(N = zero_zero_nat,bot_bot_set_nat,insert_nat(Xb,bot_bot_set_nat)) ).

% set_replicate_conv_if
tff(fact_8370_set__replicate__conv__if,axiom,
    ! [N: nat,Xb: int] :
      set_int2(replicate_int(N,Xb)) = $ite(N = zero_zero_nat,bot_bot_set_int,insert_int(Xb,bot_bot_set_int)) ).

% set_replicate_conv_if
tff(fact_8371_powr__realpow,axiom,
    ! [Xb: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),aa_nat_real(semiri5074537144036343181t_real,N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N) ) ) ).

% powr_realpow
tff(fact_8372_powr__less__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(B2),Y)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(log(B2),Xb)) ) ) ) ).

% powr_less_iff
tff(fact_8373_less__powr__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(powr_real(B2),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B2),Xb)),Y) ) ) ) ).

% less_powr_iff
tff(fact_8374_log__less__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B2),Xb)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(powr_real(B2),Y)) ) ) ) ).

% log_less_iff
tff(fact_8375_less__log__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(log(B2),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(B2),Y)),Xb) ) ) ) ).

% less_log_iff
tff(fact_8376_powr__minus__divide,axiom,
    ! [Xb: real,A2: real] : aa_real_real(powr_real(Xb),aa_real_real(uminus_uminus_real,A2)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(powr_real(Xb),A2)) ).

% powr_minus_divide
tff(fact_8377_powr__neg__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(divide_divide_real(one_one_real),Xb) ) ) ).

% powr_neg_one
tff(fact_8378_powr__mult__base,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(times_times_real(Xb),aa_real_real(powr_real(Xb),Y)) = aa_real_real(powr_real(Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_8379_powr__le__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(B2),Y)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(log(B2),Xb)) ) ) ) ).

% powr_le_iff
tff(fact_8380_le__powr__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(powr_real(B2),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B2),Xb)),Y) ) ) ) ).

% le_powr_iff
tff(fact_8381_log__le__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B2),Xb)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(powr_real(B2),Y)) ) ) ) ).

% log_le_iff
tff(fact_8382_le__log__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(log(B2),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(B2),Y)),Xb) ) ) ) ).

% le_log_iff
tff(fact_8383_ln__powr__bound,axiom,
    ! [Xb: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(divide_divide_real(aa_real_real(powr_real(Xb),A2)),A2)) ) ) ).

% ln_powr_bound
tff(fact_8384_ln__powr__bound2,axiom,
    ! [Xb: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(aa_real_real(ln_ln_real,Xb)),A2)),aa_real_real(times_times_real(aa_real_real(powr_real(A2),A2)),Xb)) ) ) ).

% ln_powr_bound2
tff(fact_8385_add__log__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( ( B2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(log(B2),Xb)) = aa_real_real(log(B2),aa_real_real(times_times_real(aa_real_real(powr_real(B2),Y)),Xb)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_8386_log__add__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( ( B2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(log(B2),Xb)),Y) = aa_real_real(log(B2),aa_real_real(times_times_real(Xb),aa_real_real(powr_real(B2),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_8387_minus__log__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( ( B2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(minus_minus_real(Y),aa_real_real(log(B2),Xb)) = aa_real_real(log(B2),aa_real_real(divide_divide_real(aa_real_real(powr_real(B2),Y)),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_8388_log__minus__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( ( B2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(minus_minus_real(aa_real_real(log(B2),Xb)),Y) = aa_real_real(log(B2),aa_real_real(times_times_real(Xb),aa_real_real(powr_real(B2),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_8389_powr__half__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(sqrt,Xb) ) ) ).

% powr_half_sqrt
tff(fact_8390_powr__neg__numeral,axiom,
    ! [Xb: real,N: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),aa_real_real(uminus_uminus_real,numeral_numeral_real(N))) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,N))) ) ) ).

% powr_neg_numeral
tff(fact_8391_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half))),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half))),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_8392_vebt__buildup_Opelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),vEBT_vebt_buildup(half))),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),vEBT_vebt_buildup(half))),vEBT_vebt_buildup(aa_nat_nat(suc,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_8393_arcosh__def,axiom,
    ! [Xb: real] : aa_real_real(arcosh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(powr_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))))) ).

% arcosh_def
tff(fact_8394_sum__gp,axiom,
    ! [Xb: complex,M: nat,N: nat] :
      groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),
        zero_zero_complex,
        $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)),aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),M)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),aa_nat_nat(suc,N)))),aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,one_one_complex),Xb))) ) ).

% sum_gp
tff(fact_8395_sum__gp,axiom,
    ! [Xb: rat,M: nat,N: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),
        zero_zero_rat,
        $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)),divide_divide_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),M),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,N))),minus_minus_rat(one_one_rat,Xb))) ) ).

% sum_gp
tff(fact_8396_sum__gp,axiom,
    ! [Xb: real,M: nat,N: nat] :
      groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M),
        zero_zero_real,
        $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,N)))),aa_real_real(minus_minus_real(one_one_real),Xb))) ) ).

% sum_gp
tff(fact_8397_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_8398_summable__complex__of__real,axiom,
    ! [Fa: fun_nat_real] :
      ( summable_complex(aTP_Lamm_agg(Fa))
    <=> summable_real(Fa) ) ).

% summable_complex_of_real
tff(fact_8399_sum_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_iv),A3) = zero_zero_nat ).

% sum.neutral_const
tff(fact_8400_sum_Oneutral__const,axiom,
    ! [A3: set_complex] : groups7754918857620584856omplex(aTP_Lamm_agh,A3) = zero_zero_complex ).

% sum.neutral_const
tff(fact_8401_sum_Oneutral__const,axiom,
    ! [A3: set_nat] : groups6591440286371151544t_real(aTP_Lamm_tz,A3) = zero_zero_real ).

% sum.neutral_const
tff(fact_8402_sum_Oneutral__const,axiom,
    ! [A3: set_int] : groups4538972089207619220nt_int(aTP_Lamm_iw,A3) = zero_zero_int ).

% sum.neutral_const
tff(fact_8403_of__nat__sum,axiom,
    ! [Fa: fun_complex_nat,A3: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(Fa,A3)) = groups7754918857620584856omplex(aTP_Lamm_agi(Fa),A3) ).

% of_nat_sum
tff(fact_8404_of__nat__sum,axiom,
    ! [Fa: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(Fa,A3)) = groups4538972089207619220nt_int(aTP_Lamm_agj(Fa),A3) ).

% of_nat_sum
tff(fact_8405_of__nat__sum,axiom,
    ! [Fa: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A3)) = groups3539618377306564664at_int(aTP_Lamm_agk(Fa),A3) ).

% of_nat_sum
tff(fact_8406_of__nat__sum,axiom,
    ! [Fa: fun_nat_nat,A3: set_nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A3)) = groups7501900531339628137nteger(aTP_Lamm_agl(Fa),A3) ).

% of_nat_sum
tff(fact_8407_of__nat__sum,axiom,
    ! [Fa: fun_nat_nat,A3: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_agm(Fa)),A3) ).

% of_nat_sum
tff(fact_8408_of__nat__sum,axiom,
    ! [Fa: fun_nat_nat,A3: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A3)) = groups6591440286371151544t_real(aTP_Lamm_mq(Fa),A3) ).

% of_nat_sum
tff(fact_8409_of__int__sum,axiom,
    ! [Fa: fun_complex_int,A3: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(Fa,A3)) = groups7754918857620584856omplex(aTP_Lamm_agn(Fa),A3) ).

% of_int_sum
tff(fact_8410_of__int__sum,axiom,
    ! [Fa: fun_nat_int,A3: set_nat] : aa_int_real(ring_1_of_int_real,groups3539618377306564664at_int(Fa,A3)) = groups6591440286371151544t_real(aTP_Lamm_ago(Fa),A3) ).

% of_int_sum
tff(fact_8411_of__int__sum,axiom,
    ! [Fa: fun_int_int,A3: set_int] : aa_int_real(ring_1_of_int_real,groups4538972089207619220nt_int(Fa,A3)) = groups8778361861064173332t_real(aTP_Lamm_agp(Fa),A3) ).

% of_int_sum
tff(fact_8412_of__int__sum,axiom,
    ! [Fa: fun_int_int,A3: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(Fa,A3)) = groups3906332499630173760nt_rat(aTP_Lamm_agq(Fa),A3) ).

% of_int_sum
tff(fact_8413_of__int__sum,axiom,
    ! [Fa: fun_int_int,A3: set_int] : ring_17408606157368542149l_num1(groups4538972089207619220nt_int(Fa,A3)) = groups8925579862173457374l_num1(aTP_Lamm_agr(Fa),A3) ).

% of_int_sum
tff(fact_8414_of__int__sum,axiom,
    ! [Fa: fun_int_int,A3: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(Fa,A3)) = groups4538972089207619220nt_int(aTP_Lamm_ags(Fa),A3) ).

% of_int_sum
tff(fact_8415_abs__sum__abs,axiom,
    ! [Fa: fun_nat_real,A3: set_nat] : aa_real_real(abs_abs_real,groups6591440286371151544t_real(aTP_Lamm_vp(Fa),A3)) = groups6591440286371151544t_real(aTP_Lamm_vp(Fa),A3) ).

% abs_sum_abs
tff(fact_8416_abs__sum__abs,axiom,
    ! [Fa: fun_int_int,A3: set_int] : aa_int_int(abs_abs_int,groups4538972089207619220nt_int(aTP_Lamm_agt(Fa),A3)) = groups4538972089207619220nt_int(aTP_Lamm_agt(Fa),A3) ).

% abs_sum_abs
tff(fact_8417_of__real__sum,axiom,
    ! [Fa: fun_complex_real,S: set_complex] : aa_real_complex(real_V4546457046886955230omplex,groups5808333547571424918x_real(Fa,S)) = groups7754918857620584856omplex(aTP_Lamm_agu(Fa),S) ).

% of_real_sum
tff(fact_8418_of__real__sum,axiom,
    ! [Fa: fun_nat_real,S: set_nat] : aa_real_complex(real_V4546457046886955230omplex,groups6591440286371151544t_real(Fa,S)) = groups2073611262835488442omplex(aTP_Lamm_agg(Fa),S) ).

% of_real_sum
tff(fact_8419_of__real__sum,axiom,
    ! [Fa: fun_nat_real,S: set_nat] : aa_real_real(real_V1803761363581548252l_real,groups6591440286371151544t_real(Fa,S)) = groups6591440286371151544t_real(aTP_Lamm_agv(Fa),S) ).

% of_real_sum
tff(fact_8420_sum_Oempty,axiom,
    ! [G: fun_real_real] : groups8097168146408367636l_real(G,bot_bot_set_real) = zero_zero_real ).

% sum.empty
tff(fact_8421_sum_Oempty,axiom,
    ! [G: fun_real_rat] : groups1300246762558778688al_rat(G,bot_bot_set_real) = zero_zero_rat ).

% sum.empty
tff(fact_8422_sum_Oempty,axiom,
    ! [G: fun_real_nat] : groups1935376822645274424al_nat(G,bot_bot_set_real) = zero_zero_nat ).

% sum.empty
tff(fact_8423_sum_Oempty,axiom,
    ! [G: fun_real_int] : groups1932886352136224148al_int(G,bot_bot_set_real) = zero_zero_int ).

% sum.empty
tff(fact_8424_sum_Oempty,axiom,
    ! [G: fun_nat_rat] : groups2906978787729119204at_rat(G,bot_bot_set_nat) = zero_zero_rat ).

% sum.empty
tff(fact_8425_sum_Oempty,axiom,
    ! [G: fun_nat_int] : groups3539618377306564664at_int(G,bot_bot_set_nat) = zero_zero_int ).

% sum.empty
tff(fact_8426_sum_Oempty,axiom,
    ! [G: fun_int_real] : groups8778361861064173332t_real(G,bot_bot_set_int) = zero_zero_real ).

% sum.empty
tff(fact_8427_sum_Oempty,axiom,
    ! [G: fun_int_rat] : groups3906332499630173760nt_rat(G,bot_bot_set_int) = zero_zero_rat ).

% sum.empty
tff(fact_8428_sum_Oempty,axiom,
    ! [G: fun_int_nat] : groups4541462559716669496nt_nat(G,bot_bot_set_int) = zero_zero_nat ).

% sum.empty
tff(fact_8429_sum_Oempty,axiom,
    ! [G: fun_nat_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),bot_bot_set_nat) = zero_zero_nat ).

% sum.empty
tff(fact_8430_sum_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_real] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( groups8778361861064173332t_real(G,A3) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_8431_sum_Oinfinite,axiom,
    ! [A3: set_complex,G: fun_complex_real] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5808333547571424918x_real(G,A3) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_8432_sum_Oinfinite,axiom,
    ! [A3: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups1270011288395367621r_real(G,A3) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_8433_sum_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_rat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( groups2906978787729119204at_rat(G,A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_8434_sum_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_rat] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( groups3906332499630173760nt_rat(G,A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_8435_sum_Oinfinite,axiom,
    ! [A3: set_complex,G: fun_complex_rat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5058264527183730370ex_rat(G,A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_8436_sum_Oinfinite,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_rat] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups6602215022474089585er_rat(G,A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_8437_sum_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_nat] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( groups4541462559716669496nt_nat(G,A3) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_8438_sum_Oinfinite,axiom,
    ! [A3: set_complex,G: fun_complex_nat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5693394587270226106ex_nat(G,A3) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_8439_sum_Oinfinite,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_nat] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups7237345082560585321er_nat(G,A3) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_8440_sum__eq__0__iff,axiom,
    ! [F4: set_int,Fa: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,F4)
     => ( ( groups4541462559716669496nt_nat(Fa,F4) = zero_zero_nat )
      <=> ! [X: int] :
            ( aa_set_int_o(member_int(X),F4)
           => ( aa_int_nat(Fa,X) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_8441_sum__eq__0__iff,axiom,
    ! [F4: set_complex,Fa: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F4)
     => ( ( groups5693394587270226106ex_nat(Fa,F4) = zero_zero_nat )
      <=> ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),F4)
           => ( aa_complex_nat(Fa,X) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_8442_sum__eq__0__iff,axiom,
    ! [F4: set_Code_integer,Fa: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F4)
     => ( ( groups7237345082560585321er_nat(Fa,F4) = zero_zero_nat )
      <=> ! [X: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X),F4)
           => ( aa_Code_integer_nat(Fa,X) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_8443_sum__eq__0__iff,axiom,
    ! [F4: set_nat,Fa: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,F4)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(Fa),F4) = zero_zero_nat )
      <=> ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),F4)
           => ( aa_nat_nat(Fa,X) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_8444_of__real__0,axiom,
    aa_real_real(real_V1803761363581548252l_real,zero_zero_real) = zero_zero_real ).

% of_real_0
tff(fact_8445_of__real__0,axiom,
    aa_real_complex(real_V4546457046886955230omplex,zero_zero_real) = zero_zero_complex ).

% of_real_0
tff(fact_8446_of__real__eq__0__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_8447_of__real__eq__0__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,Xb) = zero_zero_complex )
    <=> ( Xb = zero_zero_real ) ) ).

% of_real_eq_0_iff
tff(fact_8448_of__real__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,Xb) = one_one_real )
    <=> ( Xb = one_one_real ) ) ).

% of_real_eq_1_iff
tff(fact_8449_of__real__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,Xb) = one_one_complex )
    <=> ( Xb = one_one_real ) ) ).

% of_real_eq_1_iff
tff(fact_8450_of__real__1,axiom,
    aa_real_real(real_V1803761363581548252l_real,one_one_real) = one_one_real ).

% of_real_1
tff(fact_8451_of__real__1,axiom,
    aa_real_complex(real_V4546457046886955230omplex,one_one_real) = one_one_complex ).

% of_real_1
tff(fact_8452_of__real__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,numeral_numeral_real(W)) = numeral_numeral_real(W) ).

% of_real_numeral
tff(fact_8453_of__real__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,numeral_numeral_real(W)) = numera6690914467698888265omplex(W) ).

% of_real_numeral
tff(fact_8454_of__real__divide,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(Xb),Y)) = aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_divide
tff(fact_8455_of__real__divide,axiom,
    ! [Xb: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(divide_divide_real(Xb),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_divide
tff(fact_8456_of__real__power,axiom,
    ! [Xb: real,N: nat] : aa_real_real(real_V1803761363581548252l_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V1803761363581548252l_real,Xb)),N) ).

% of_real_power
tff(fact_8457_of__real__power,axiom,
    ! [Xb: real,N: nat] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_real_complex(real_V4546457046886955230omplex,Xb)),N) ).

% of_real_power
tff(fact_8458_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_8459_sum_Odelta_H,axiom,
    ! [S3: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_agw(A2),B2),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),S3),aa_VEBT_VEBT_real(B2,A2),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_8460_sum_Odelta_H,axiom,
    ! [S3: set_real,A2: real,B2: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_agx(A2),B2),S3) = $ite(aa_set_real_o(member_real(A2),S3),aa_real_real(B2,A2),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_8461_sum_Odelta_H,axiom,
    ! [S3: set_int,A2: int,B2: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_agy(A2),B2),S3) = $ite(aa_set_int_o(member_int(A2),S3),aa_int_real(B2,A2),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_8462_sum_Odelta_H,axiom,
    ! [S3: set_complex,A2: complex,B2: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_agz(A2),B2),S3) = $ite(aa_set_complex_o(member_complex(A2),S3),aa_complex_real(B2,A2),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_8463_sum_Odelta_H,axiom,
    ! [S3: set_Code_integer,A2: code_integer,B2: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( groups1270011288395367621r_real(aa_fun6341089502340155082r_real(aTP_Lamm_aha(A2),B2),S3) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A2),S3),aa_Code_integer_real(B2,A2),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_8464_sum_Odelta_H,axiom,
    ! [S3: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_ahb(A2),B2),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),S3),aa_VEBT_VEBT_rat(B2,A2),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_8465_sum_Odelta_H,axiom,
    ! [S3: set_real,A2: real,B2: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_ahc(A2),B2),S3) = $ite(aa_set_real_o(member_real(A2),S3),aa_real_rat(B2,A2),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_8466_sum_Odelta_H,axiom,
    ! [S3: set_nat,A2: nat,B2: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ahd(A2),B2),S3) = $ite(aa_set_nat_o(member_nat(A2),S3),aa_nat_rat(B2,A2),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_8467_sum_Odelta_H,axiom,
    ! [S3: set_int,A2: int,B2: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ahe(A2),B2),S3) = $ite(aa_set_int_o(member_int(A2),S3),aa_int_rat(B2,A2),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_8468_sum_Odelta_H,axiom,
    ! [S3: set_complex,A2: complex,B2: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_ahf(A2),B2),S3) = $ite(aa_set_complex_o(member_complex(A2),S3),aa_complex_rat(B2,A2),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_8469_sum_Odelta,axiom,
    ! [S3: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aTP_Lamm_ahg(A2),B2),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),S3),aa_VEBT_VEBT_real(B2,A2),zero_zero_real) ) ) ).

% sum.delta
tff(fact_8470_sum_Odelta,axiom,
    ! [S3: set_real,A2: real,B2: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ahh(A2),B2),S3) = $ite(aa_set_real_o(member_real(A2),S3),aa_real_real(B2,A2),zero_zero_real) ) ) ).

% sum.delta
tff(fact_8471_sum_Odelta,axiom,
    ! [S3: set_int,A2: int,B2: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ahi(A2),B2),S3) = $ite(aa_set_int_o(member_int(A2),S3),aa_int_real(B2,A2),zero_zero_real) ) ) ).

% sum.delta
tff(fact_8472_sum_Odelta,axiom,
    ! [S3: set_complex,A2: complex,B2: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ahj(A2),B2),S3) = $ite(aa_set_complex_o(member_complex(A2),S3),aa_complex_real(B2,A2),zero_zero_real) ) ) ).

% sum.delta
tff(fact_8473_sum_Odelta,axiom,
    ! [S3: set_Code_integer,A2: code_integer,B2: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( groups1270011288395367621r_real(aa_fun6341089502340155082r_real(aTP_Lamm_ahk(A2),B2),S3) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A2),S3),aa_Code_integer_real(B2,A2),zero_zero_real) ) ) ).

% sum.delta
tff(fact_8474_sum_Odelta,axiom,
    ! [S3: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_ahl(A2),B2),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),S3),aa_VEBT_VEBT_rat(B2,A2),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_8475_sum_Odelta,axiom,
    ! [S3: set_real,A2: real,B2: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S3)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_ahm(A2),B2),S3) = $ite(aa_set_real_o(member_real(A2),S3),aa_real_rat(B2,A2),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_8476_sum_Odelta,axiom,
    ! [S3: set_nat,A2: nat,B2: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ahn(A2),B2),S3) = $ite(aa_set_nat_o(member_nat(A2),S3),aa_nat_rat(B2,A2),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_8477_sum_Odelta,axiom,
    ! [S3: set_int,A2: int,B2: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_aho(A2),B2),S3) = $ite(aa_set_int_o(member_int(A2),S3),aa_int_rat(B2,A2),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_8478_sum_Odelta,axiom,
    ! [S3: set_complex,A2: complex,B2: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_ahp(A2),B2),S3) = $ite(aa_set_complex_o(member_complex(A2),S3),aa_complex_rat(B2,A2),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_8479_sum__abs,axiom,
    ! [Fa: fun_nat_real,A3: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,groups6591440286371151544t_real(Fa,A3))),groups6591440286371151544t_real(aTP_Lamm_vp(Fa),A3)) ).

% sum_abs
tff(fact_8480_sum__abs,axiom,
    ! [Fa: fun_int_int,A3: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(Fa,A3))),groups4538972089207619220nt_int(aTP_Lamm_agt(Fa),A3)) ).

% sum_abs
tff(fact_8481_sin__of__real__pi,axiom,
    aa_real_real(sin_real,aa_real_real(real_V1803761363581548252l_real,pi)) = zero_zero_real ).

% sin_of_real_pi
tff(fact_8482_sin__of__real__pi,axiom,
    aa_complex_complex(sin_complex,aa_real_complex(real_V4546457046886955230omplex,pi)) = zero_zero_complex ).

% sin_of_real_pi
tff(fact_8483_sum__abs__ge__zero,axiom,
    ! [Fa: fun_nat_real,A3: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups6591440286371151544t_real(aTP_Lamm_vp(Fa),A3)) ).

% sum_abs_ge_zero
tff(fact_8484_sum__abs__ge__zero,axiom,
    ! [Fa: fun_int_int,A3: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_agt(Fa),A3)) ).

% sum_abs_ge_zero
tff(fact_8485_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) ).

% of_real_neg_numeral
tff(fact_8486_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(uminus_uminus_real,numeral_numeral_real(W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% of_real_neg_numeral
tff(fact_8487_cos__of__real__pi,axiom,
    aa_real_real(cos_real,aa_real_real(real_V1803761363581548252l_real,pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_of_real_pi
tff(fact_8488_cos__of__real__pi,axiom,
    aa_complex_complex(cos_complex,aa_real_complex(real_V4546457046886955230omplex,pi)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% cos_of_real_pi
tff(fact_8489_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_na2984781812408219459l_num1,M: nat,N: nat] :
      groups2996710295995929986l_num1(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),zero_z3563351764282998399l_num1,plus_p361126936061061375l_num1(groups2996710295995929986l_num1(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)),aa_nat5599461682457136004l_num1(G,aa_nat_nat(suc,N)))) ).

% sum.cl_ivl_Suc
tff(fact_8490_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,N: nat] :
      groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),zero_zero_rat,plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)),aa_nat_rat(G,aa_nat_nat(suc,N)))) ).

% sum.cl_ivl_Suc
tff(fact_8491_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,N: nat] :
      groups3539618377306564664at_int(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))),aa_nat_int(G,aa_nat_nat(suc,N)))) ).

% sum.cl_ivl_Suc
tff(fact_8492_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))),aa_nat_nat(G,aa_nat_nat(suc,N)))) ).

% sum.cl_ivl_Suc
tff(fact_8493_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] :
      groups6591440286371151544t_real(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,N)),M),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))),aa_nat_real(G,aa_nat_nat(suc,N)))) ).

% sum.cl_ivl_Suc
tff(fact_8494_sum__zero__power,axiom,
    ! [C: fun_nat_complex,A3: set_nat] :
      groups2073611262835488442omplex(aTP_Lamm_tu(C),A3) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A3)
        & aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
        aa_nat_complex(C,zero_zero_nat),
        zero_zero_complex ) ).

% sum_zero_power
tff(fact_8495_sum__zero__power,axiom,
    ! [C: fun_nat_rat,A3: set_nat] :
      groups2906978787729119204at_rat(aTP_Lamm_ahq(C),A3) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A3)
        & aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
        aa_nat_rat(C,zero_zero_nat),
        zero_zero_rat ) ).

% sum_zero_power
tff(fact_8496_sum__zero__power,axiom,
    ! [C: fun_nat_real,A3: set_nat] :
      groups6591440286371151544t_real(aTP_Lamm_tv(C),A3) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A3)
        & aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
        aa_nat_real(C,zero_zero_nat),
        zero_zero_real ) ).

% sum_zero_power
tff(fact_8497_norm__of__real__add1,axiom,
    ! [Xb: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,Xb)),one_one_real)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) ).

% norm_of_real_add1
tff(fact_8498_norm__of__real__add1,axiom,
    ! [Xb: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,Xb)),one_one_complex)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) ).

% norm_of_real_add1
tff(fact_8499_norm__of__real__addn,axiom,
    ! [Xb: real,B2: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,Xb)),numeral_numeral_real(B2))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),numeral_numeral_real(B2))) ).

% norm_of_real_addn
tff(fact_8500_norm__of__real__addn,axiom,
    ! [Xb: real,B2: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,Xb)),numera6690914467698888265omplex(B2))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),numeral_numeral_real(B2))) ).

% norm_of_real_addn
tff(fact_8501_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_8502_sum__zero__power_H,axiom,
    ! [C: fun_nat_complex,D: fun_nat_complex,A3: set_nat] :
      groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ahr(C),D),A3) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A3)
        & aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
        aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(C,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)),
        zero_zero_complex ) ).

% sum_zero_power'
tff(fact_8503_sum__zero__power_H,axiom,
    ! [C: fun_nat_rat,D: fun_nat_rat,A3: set_nat] :
      groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ahs(C),D),A3) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A3)
        & aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
        divide_divide_rat(aa_nat_rat(C,zero_zero_nat),aa_nat_rat(D,zero_zero_nat)),
        zero_zero_rat ) ).

% sum_zero_power'
tff(fact_8504_sum__zero__power_H,axiom,
    ! [C: fun_nat_real,D: fun_nat_real,A3: set_nat] :
      groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_aht(C),D),A3) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A3)
        & aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
        aa_real_real(divide_divide_real(aa_nat_real(C,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)),
        zero_zero_real ) ).

% sum_zero_power'
tff(fact_8505_cos__of__real__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_of_real_pi_half
tff(fact_8506_cos__of__real__pi__half,axiom,
    aa_complex_complex(cos_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = zero_zero_complex ).

% cos_of_real_pi_half
tff(fact_8507_sin__of__real__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))) = one_one_real ).

% sin_of_real_pi_half
tff(fact_8508_sin__of__real__pi__half,axiom,
    aa_complex_complex(sin_complex,aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = one_one_complex ).

% sin_of_real_pi_half
tff(fact_8509_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_real,A3: set_VEBT_VEBT] :
      ( ( groups2240296850493347238T_real(G,A3) != zero_zero_real )
     => ~ ! [A: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A3)
           => ( aa_VEBT_VEBT_real(G,A) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8510_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A3: set_int] :
      ( ( groups8778361861064173332t_real(G,A3) != zero_zero_real )
     => ~ ! [A: int] :
            ( aa_set_int_o(member_int(A),A3)
           => ( aa_int_real(G,A) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8511_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A3: set_real] :
      ( ( groups8097168146408367636l_real(G,A3) != zero_zero_real )
     => ~ ! [A: real] :
            ( aa_set_real_o(member_real(A),A3)
           => ( aa_real_real(G,A) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8512_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A3: set_nat] :
      ( ( groups2906978787729119204at_rat(G,A3) != zero_zero_rat )
     => ~ ! [A: nat] :
            ( aa_set_nat_o(member_nat(A),A3)
           => ( aa_nat_rat(G,A) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8513_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_rat,A3: set_VEBT_VEBT] :
      ( ( groups136491112297645522BT_rat(G,A3) != zero_zero_rat )
     => ~ ! [A: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A3)
           => ( aa_VEBT_VEBT_rat(G,A) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8514_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A3: set_int] :
      ( ( groups3906332499630173760nt_rat(G,A3) != zero_zero_rat )
     => ~ ! [A: int] :
            ( aa_set_int_o(member_int(A),A3)
           => ( aa_int_rat(G,A) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8515_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A3: set_real] :
      ( ( groups1300246762558778688al_rat(G,A3) != zero_zero_rat )
     => ~ ! [A: real] :
            ( aa_set_real_o(member_real(A),A3)
           => ( aa_real_rat(G,A) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8516_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_VEBT_VEBT_nat,A3: set_VEBT_VEBT] :
      ( ( groups771621172384141258BT_nat(G,A3) != zero_zero_nat )
     => ~ ! [A: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),A3)
           => ( aa_VEBT_VEBT_nat(G,A) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8517_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_nat,A3: set_int] :
      ( ( groups4541462559716669496nt_nat(G,A3) != zero_zero_nat )
     => ~ ! [A: int] :
            ( aa_set_int_o(member_int(A),A3)
           => ( aa_int_nat(G,A) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8518_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_nat,A3: set_real] :
      ( ( groups1935376822645274424al_nat(G,A3) != zero_zero_nat )
     => ~ ! [A: real] :
            ( aa_set_real_o(member_real(A),A3)
           => ( aa_real_nat(G,A) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_8519_sum_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_nat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A3)
         => ( aa_nat_nat(G,X2) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_8520_sum_Oneutral,axiom,
    ! [A3: set_complex,G: fun_complex_complex] :
      ( ! [X2: complex] :
          ( aa_set_complex_o(member_complex(X2),A3)
         => ( aa_complex_complex(G,X2) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A3) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_8521_sum_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_real] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A3)
         => ( aa_nat_real(G,X2) = zero_zero_real ) )
     => ( groups6591440286371151544t_real(G,A3) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_8522_sum_Oneutral,axiom,
    ! [A3: set_int,G: fun_int_int] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A3)
         => ( aa_int_int(G,X2) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G,A3) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_8523_sum_Oswap,axiom,
    ! [G: fun_nat_fun_nat_nat,B5: set_nat,A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_ahu(G),B5)),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aTP_Lamm_ahw(G),A3)),B5) ).

% sum.swap
tff(fact_8524_sum_Oswap,axiom,
    ! [G: fun_co7673397065115134084omplex,B5: set_complex,A3: set_complex] : groups7754918857620584856omplex(aa_set7743095458595710203omplex(aTP_Lamm_ahx(G),B5),A3) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aTP_Lamm_ahz(G),A3),B5) ).

% sum.swap
tff(fact_8525_sum_Oswap,axiom,
    ! [G: fun_nat_fun_nat_real,B5: set_nat,A3: set_nat] : groups6591440286371151544t_real(aa_set1166323925324055613t_real(aTP_Lamm_aia(G),B5),A3) = groups6591440286371151544t_real(aa_set1166323925324055613t_real(aTP_Lamm_aic(G),A3),B5) ).

% sum.swap
tff(fact_8526_sum_Oswap,axiom,
    ! [G: fun_int_fun_int_int,B5: set_int,A3: set_int] : groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_aid(G),B5),A3) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aTP_Lamm_aif(G),A3),B5) ).

% sum.swap
tff(fact_8527_norm__sum,axiom,
    ! [Fa: fun_nat_complex,A3: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups2073611262835488442omplex(Fa,A3))),groups6591440286371151544t_real(aTP_Lamm_uy(Fa),A3)) ).

% norm_sum
tff(fact_8528_norm__sum,axiom,
    ! [Fa: fun_complex_complex,A3: set_complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(Fa,A3))),groups5808333547571424918x_real(aTP_Lamm_aig(Fa),A3)) ).

% norm_sum
tff(fact_8529_norm__sum,axiom,
    ! [Fa: fun_nat_real,A3: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,groups6591440286371151544t_real(Fa,A3))),groups6591440286371151544t_real(aTP_Lamm_ux(Fa),A3)) ).

% norm_sum
tff(fact_8530_sum__mono,axiom,
    ! [K5: set_nat,Fa: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,I2)),aa_nat_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(Fa,K5)),groups2906978787729119204at_rat(G,K5)) ) ).

% sum_mono
tff(fact_8531_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(Fa,I2)),aa_VEBT_VEBT_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups136491112297645522BT_rat(Fa,K5)),groups136491112297645522BT_rat(G,K5)) ) ).

% sum_mono
tff(fact_8532_sum__mono,axiom,
    ! [K5: set_int,Fa: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,I2)),aa_int_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(Fa,K5)),groups3906332499630173760nt_rat(G,K5)) ) ).

% sum_mono
tff(fact_8533_sum__mono,axiom,
    ! [K5: set_real,Fa: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),K5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(Fa,I2)),aa_real_rat(G,I2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(Fa,K5)),groups1300246762558778688al_rat(G,K5)) ) ).

% sum_mono
tff(fact_8534_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(Fa,I2)),aa_VEBT_VEBT_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups771621172384141258BT_nat(Fa,K5)),groups771621172384141258BT_nat(G,K5)) ) ).

% sum_mono
tff(fact_8535_sum__mono,axiom,
    ! [K5: set_int,Fa: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Fa,I2)),aa_int_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(Fa,K5)),groups4541462559716669496nt_nat(G,K5)) ) ).

% sum_mono
tff(fact_8536_sum__mono,axiom,
    ! [K5: set_real,Fa: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),K5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(Fa,I2)),aa_real_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(Fa,K5)),groups1935376822645274424al_nat(G,K5)) ) ).

% sum_mono
tff(fact_8537_sum__mono,axiom,
    ! [K5: set_nat,Fa: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(Fa,I2)),aa_nat_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(Fa,K5)),groups3539618377306564664at_int(G,K5)) ) ).

% sum_mono
tff(fact_8538_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_int,G: fun_VEBT_VEBT_int] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_VEBT_VEBT_int(Fa,I2)),aa_VEBT_VEBT_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups769130701875090982BT_int(Fa,K5)),groups769130701875090982BT_int(G,K5)) ) ).

% sum_mono
tff(fact_8539_sum__mono,axiom,
    ! [K5: set_real,Fa: fun_real_int,G: fun_real_int] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),K5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(Fa,I2)),aa_real_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(Fa,K5)),groups1932886352136224148al_int(G,K5)) ) ).

% sum_mono
tff(fact_8540_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_vd(G),H)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A3)) ).

% sum.distrib
tff(fact_8541_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A3: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_aih(G),H),A3) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,groups7754918857620584856omplex(G,A3)),groups7754918857620584856omplex(H,A3)) ).

% sum.distrib
tff(fact_8542_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A3: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_vc(G),H),A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,A3)),groups6591440286371151544t_real(H,A3)) ).

% sum.distrib
tff(fact_8543_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A3: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_aii(G),H),A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A3)),groups4538972089207619220nt_int(H,A3)) ).

% sum.distrib
tff(fact_8544_sum__product,axiom,
    ! [Fa: fun_nat_nat,A3: set_nat,G: fun_nat_nat,B5: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B5)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_aik(Fa),G),B5)),A3) ).

% sum_product
tff(fact_8545_sum__product,axiom,
    ! [Fa: fun_complex_complex,A3: set_complex,G: fun_complex_complex,B5: set_complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,groups7754918857620584856omplex(Fa,A3)),groups7754918857620584856omplex(G,B5)) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_aim(Fa),G),B5),A3) ).

% sum_product
tff(fact_8546_sum__product,axiom,
    ! [Fa: fun_nat_real,A3: set_nat,G: fun_nat_real,B5: set_nat] : aa_real_real(times_times_real(groups6591440286371151544t_real(Fa,A3)),groups6591440286371151544t_real(G,B5)) = groups6591440286371151544t_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_aio(Fa),G),B5),A3) ).

% sum_product
tff(fact_8547_sum__product,axiom,
    ! [Fa: fun_int_int,A3: set_int,G: fun_int_int,B5: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(Fa,A3)),groups4538972089207619220nt_int(G,B5)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_aiq(Fa),G),B5),A3) ).

% sum_product
tff(fact_8548_sum__distrib__right,axiom,
    ! [Fa: fun_nat_nat,A3: set_nat,R: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A3)),R) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_air(Fa),R)),A3) ).

% sum_distrib_right
tff(fact_8549_sum__distrib__right,axiom,
    ! [Fa: fun_complex_complex,A3: set_complex,R: complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,groups7754918857620584856omplex(Fa,A3)),R) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_ais(Fa),R),A3) ).

% sum_distrib_right
tff(fact_8550_sum__distrib__right,axiom,
    ! [Fa: fun_nat_real,A3: set_nat,R: real] : aa_real_real(times_times_real(groups6591440286371151544t_real(Fa,A3)),R) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vg(Fa),R),A3) ).

% sum_distrib_right
tff(fact_8551_sum__distrib__right,axiom,
    ! [Fa: fun_int_int,A3: set_int,R: int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(Fa,A3)),R) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_ait(Fa),R),A3) ).

% sum_distrib_right
tff(fact_8552_sum__distrib__left,axiom,
    ! [R: nat,Fa: fun_nat_nat,A3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aiu(R),Fa)),A3) ).

% sum_distrib_left
tff(fact_8553_sum__distrib__left,axiom,
    ! [R: complex,Fa: fun_complex_complex,A3: set_complex] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,R),groups7754918857620584856omplex(Fa,A3)) = groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_aiv(R),Fa),A3) ).

% sum_distrib_left
tff(fact_8554_sum__distrib__left,axiom,
    ! [R: real,Fa: fun_nat_real,A3: set_nat] : aa_real_real(times_times_real(R),groups6591440286371151544t_real(Fa,A3)) = groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_uu(R),Fa),A3) ).

% sum_distrib_left
tff(fact_8555_sum__distrib__left,axiom,
    ! [R: int,Fa: fun_int_int,A3: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,R),groups4538972089207619220nt_int(Fa,A3)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_aiw(R),Fa),A3) ).

% sum_distrib_left
tff(fact_8556_sum__subtractf,axiom,
    ! [Fa: fun_complex_complex,G: fun_complex_complex,A3: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_aix(Fa),G),A3) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,groups7754918857620584856omplex(Fa,A3)),groups7754918857620584856omplex(G,A3)) ).

% sum_subtractf
tff(fact_8557_sum__subtractf,axiom,
    ! [Fa: fun_nat_real,G: fun_nat_real,A3: set_nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_vk(Fa),G),A3) = aa_real_real(minus_minus_real(groups6591440286371151544t_real(Fa,A3)),groups6591440286371151544t_real(G,A3)) ).

% sum_subtractf
tff(fact_8558_sum__subtractf,axiom,
    ! [Fa: fun_int_int,G: fun_int_int,A3: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_aiy(Fa),G),A3) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(Fa,A3)),groups4538972089207619220nt_int(G,A3)) ).

% sum_subtractf
tff(fact_8559_sum__divide__distrib,axiom,
    ! [Fa: fun_complex_complex,A3: set_complex,R: complex] : aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,groups7754918857620584856omplex(Fa,A3)),R) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_aiz(Fa),R),A3) ).

% sum_divide_distrib
tff(fact_8560_sum__divide__distrib,axiom,
    ! [Fa: fun_nat_real,A3: set_nat,R: real] : aa_real_real(divide_divide_real(groups6591440286371151544t_real(Fa,A3)),R) = groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_uw(Fa),R),A3) ).

% sum_divide_distrib
tff(fact_8561_sum__negf,axiom,
    ! [Fa: fun_complex_complex,A3: set_complex] : groups7754918857620584856omplex(aTP_Lamm_aja(Fa),A3) = aa_complex_complex(uminus1482373934393186551omplex,groups7754918857620584856omplex(Fa,A3)) ).

% sum_negf
tff(fact_8562_sum__negf,axiom,
    ! [Fa: fun_nat_real,A3: set_nat] : groups6591440286371151544t_real(aTP_Lamm_vo(Fa),A3) = aa_real_real(uminus_uminus_real,groups6591440286371151544t_real(Fa,A3)) ).

% sum_negf
tff(fact_8563_sum__negf,axiom,
    ! [Fa: fun_int_int,A3: set_int] : groups4538972089207619220nt_int(aTP_Lamm_ajb(Fa),A3) = aa_int_int(uminus_uminus_int,groups4538972089207619220nt_int(Fa,A3)) ).

% sum_negf
tff(fact_8564_sum_Oswap__restrict,axiom,
    ! [A3: set_VEBT_VEBT,B5: set_nat,G: fun_VE597728521808591204at_nat,R4: fun_VE6480751511521678896_nat_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
     => ( aa_set_nat_o(finite_finite_nat,B5)
       => ( groups771621172384141258BT_nat(aa_fun3621025987637128233BT_nat(aa_fun1603736613815373464BT_nat(aTP_Lamm_ajd(B5),G),R4),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun6249118799011721995at_nat(aa_fun2705500642357200114at_nat(aTP_Lamm_ajf(A3),G),R4)),B5) ) ) ) ).

% sum.swap_restrict
tff(fact_8565_sum_Oswap__restrict,axiom,
    ! [A3: set_real,B5: set_nat,G: fun_real_fun_nat_nat,R4: fun_real_fun_nat_o] :
      ( aa_set_real_o(finite_finite_real,A3)
     => ( aa_set_nat_o(finite_finite_nat,B5)
       => ( groups1935376822645274424al_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_ajh(B5),G),R4),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_ajj(A3),G),R4)),B5) ) ) ) ).

% sum.swap_restrict
tff(fact_8566_sum_Oswap__restrict,axiom,
    ! [A3: set_int,B5: set_nat,G: fun_int_fun_nat_nat,R4: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_nat_o(finite_finite_nat,B5)
       => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_ajl(B5),G),R4),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_ajo(A3),G),R4)),B5) ) ) ) ).

% sum.swap_restrict
tff(fact_8567_sum_Oswap__restrict,axiom,
    ! [A3: set_complex,B5: set_nat,G: fun_co636942507509095880at_nat,R4: fun_co5676208753217392844_nat_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_nat_o(finite_finite_nat,B5)
       => ( groups5693394587270226106ex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_ajq(B5),G),R4),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_ajt(A3),G),R4)),B5) ) ) ) ).

% sum.swap_restrict
tff(fact_8568_sum_Oswap__restrict,axiom,
    ! [A3: set_Code_integer,B5: set_nat,G: fun_Co3312552843764780407at_nat,R4: fun_Co1375048237525739357_nat_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set_nat_o(finite_finite_nat,B5)
       => ( groups7237345082560585321er_nat(aa_fun6683192525214198859er_nat(aa_fun2143651634668453157er_nat(aTP_Lamm_ajv(B5),G),R4),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7456160887942273982at_nat(aa_fun4417447297828429592at_nat(aTP_Lamm_ajy(A3),G),R4)),B5) ) ) ) ).

% sum.swap_restrict
tff(fact_8569_sum_Oswap__restrict,axiom,
    ! [A3: set_VEBT_VEBT,B5: set_complex,G: fun_VE2006399705683203104omplex,R4: fun_VE7631951351884037842plex_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B5)
       => ( groups1794756597179926696omplex(aa_fun1953711888481404901omplex(aa_fun5829985485761010584omplex(aTP_Lamm_aka(B5),G),R4),A3) = groups7754918857620584856omplex(aa_fun2592672125884331625omplex(aa_fun8556514703313836300omplex(aTP_Lamm_akc(A3),G),R4),B5) ) ) ) ).

% sum.swap_restrict
tff(fact_8570_sum_Oswap__restrict,axiom,
    ! [A3: set_real,B5: set_complex,G: fun_re7208826350674946562omplex,R4: fun_re7319081385389557872plex_o] :
      ( aa_set_real_o(finite_finite_real,A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B5)
       => ( groups5754745047067104278omplex(aa_fun401690246322682763omplex(aa_fun7501666206714588442omplex(aTP_Lamm_ake(B5),G),R4),A3) = groups7754918857620584856omplex(aa_fun9084337531551611917omplex(aa_fun6739043078600750748omplex(aTP_Lamm_akg(A3),G),R4),B5) ) ) ) ).

% sum.swap_restrict
tff(fact_8571_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B5: set_complex,G: fun_na1591161823130732966omplex,R4: fun_na1554462459066888140plex_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B5)
       => ( groups2073611262835488442omplex(aa_fun4713049822795992587omplex(aa_fun3479084698764006518omplex(aTP_Lamm_aki(B5),G),R4),A3) = groups7754918857620584856omplex(aa_fun1423577888295648489omplex(aa_fun601428455815558228omplex(aTP_Lamm_akl(A3),G),R4),B5) ) ) ) ).

% sum.swap_restrict
tff(fact_8572_sum_Oswap__restrict,axiom,
    ! [A3: set_int,B5: set_complex,G: fun_in1307110748760623106omplex,R4: fun_in5065666251602127472plex_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B5)
       => ( groups3049146728041665814omplex(aa_fun3141780957025892107omplex(aa_fun323909820831547930omplex(aTP_Lamm_akn(B5),G),R4),A3) = groups7754918857620584856omplex(aa_fun7450476157521870733omplex(aa_fun6156949752960786844omplex(aTP_Lamm_akq(A3),G),R4),B5) ) ) ) ).

% sum.swap_restrict
tff(fact_8573_sum_Oswap__restrict,axiom,
    ! [A3: set_Code_integer,B5: set_complex,G: fun_Co8905303033605938227omplex,R4: fun_Co8975588572093281791plex_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B5)
       => ( groups8024822376189712711omplex(aa_fun9008897831946212043omplex(aa_fun8319717443755388649omplex(aTP_Lamm_aks(B5),G),R4),A3) = groups7754918857620584856omplex(aa_fun7787660480616508700omplex(aa_fun963654016877434554omplex(aTP_Lamm_akv(A3),G),R4),B5) ) ) ) ).

% sum.swap_restrict
tff(fact_8574_mod__sum__eq,axiom,
    ! [Fa: fun_nat_nat,A2: nat,A3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_akw(Fa),A2)),A3)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A3)),A2) ).

% mod_sum_eq
tff(fact_8575_mod__sum__eq,axiom,
    ! [Fa: fun_int_int,A2: int,A3: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_akx(Fa),A2),A3)),A2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(Fa,A3)),A2) ).

% mod_sum_eq
tff(fact_8576_summable__sum,axiom,
    ! [I5: set_VEBT_VEBT,Fa: fun_VE2989161857680785728t_real] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
         => summable_real(aa_VEB3736018289492012273t_real(Fa,I2)) )
     => summable_real(aa_fun6147343916763992189t_real(aTP_Lamm_akz(I5),Fa)) ) ).

% summable_sum
tff(fact_8577_summable__sum,axiom,
    ! [I5: set_int,Fa: fun_int_fun_nat_real] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I5)
         => summable_real(aa_int_fun_nat_real(Fa,I2)) )
     => summable_real(aa_fun2842512154077127769t_real(aTP_Lamm_alb(I5),Fa)) ) ).

% summable_sum
tff(fact_8578_summable__sum,axiom,
    ! [I5: set_real,Fa: fun_re4238438383274954274t_real] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),I5)
         => summable_real(aa_real_fun_nat_real(Fa,I2)) )
     => summable_real(aa_fun628238740774635481t_real(aTP_Lamm_ald(I5),Fa)) ) ).

% summable_sum
tff(fact_8579_summable__sum,axiom,
    ! [I5: set_nat,Fa: fun_na1142814060743658952omplex] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => summable_complex(aa_nat6454746962644518025omplex(Fa,I2)) )
     => summable_complex(aa_fun4886358607025424513omplex(aTP_Lamm_alf(I5),Fa)) ) ).

% summable_sum
tff(fact_8580_summable__sum,axiom,
    ! [I5: set_VEBT_VEBT,Fa: fun_VE6440932821983644738omplex] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
         => summable_complex(aa_VEB6050840761258413811omplex(Fa,I2)) )
     => summable_complex(aa_fun3960763739589619457omplex(aTP_Lamm_alh(I5),Fa)) ) ).

% summable_sum
tff(fact_8581_summable__sum,axiom,
    ! [I5: set_int,Fa: fun_in8056368165505483812omplex] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I5)
         => summable_complex(aa_int4144929030551567077omplex(Fa,I2)) )
     => summable_complex(aa_fun3543561798013398749omplex(aTP_Lamm_alj(I5),Fa)) ) ).

% summable_sum
tff(fact_8582_summable__sum,axiom,
    ! [I5: set_real,Fa: fun_re2084383175013312036omplex] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),I5)
         => summable_complex(aa_rea1531280491672885477omplex(Fa,I2)) )
     => summable_complex(aa_fun7632768522510439005omplex(aTP_Lamm_all(I5),Fa)) ) ).

% summable_sum
tff(fact_8583_summable__sum,axiom,
    ! [I5: set_nat,Fa: fun_nat_fun_nat_nat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => summable_nat(aa_nat_fun_nat_nat(Fa,I2)) )
     => summable_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_alm(I5),Fa)) ) ).

% summable_sum
tff(fact_8584_summable__sum,axiom,
    ! [I5: set_complex,Fa: fun_co3258106519541737638omplex] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I5)
         => summable_complex(aa_com8181001469867462503omplex(Fa,I2)) )
     => summable_complex(aa_fun5003373910569599327omplex(aTP_Lamm_alo(I5),Fa)) ) ).

% summable_sum
tff(fact_8585_summable__sum,axiom,
    ! [I5: set_nat,Fa: fun_nat_fun_nat_real] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => summable_real(aa_nat_fun_nat_real(Fa,I2)) )
     => summable_real(aa_fun7039663905532832253t_real(aTP_Lamm_alp(I5),Fa)) ) ).

% summable_sum
tff(fact_8586_sum__nonpos,axiom,
    ! [A3: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_real] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(Fa,X2)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2240296850493347238T_real(Fa,A3)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_8587_sum__nonpos,axiom,
    ! [A3: set_int,Fa: fun_int_real] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(Fa,X2)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(Fa,A3)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_8588_sum__nonpos,axiom,
    ! [A3: set_real,Fa: fun_real_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(Fa,X2)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(Fa,A3)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_8589_sum__nonpos,axiom,
    ! [A3: set_nat,Fa: fun_nat_rat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(Fa,A3)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_8590_sum__nonpos,axiom,
    ! [A3: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_rat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(Fa,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups136491112297645522BT_rat(Fa,A3)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_8591_sum__nonpos,axiom,
    ! [A3: set_int,Fa: fun_int_rat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(Fa,A3)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_8592_sum__nonpos,axiom,
    ! [A3: set_real,Fa: fun_real_rat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(Fa,X2)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(Fa,A3)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_8593_sum__nonpos,axiom,
    ! [A3: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(Fa,X2)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups771621172384141258BT_nat(Fa,A3)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_8594_sum__nonpos,axiom,
    ! [A3: set_int,Fa: fun_int_nat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Fa,X2)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(Fa,A3)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_8595_sum__nonpos,axiom,
    ! [A3: set_real,Fa: fun_real_nat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(Fa,X2)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(Fa,A3)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_8596_sum__nonneg,axiom,
    ! [A3: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_real] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(Fa,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups2240296850493347238T_real(Fa,A3)) ) ).

% sum_nonneg
tff(fact_8597_sum__nonneg,axiom,
    ! [A3: set_int,Fa: fun_int_real] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Fa,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(Fa,A3)) ) ).

% sum_nonneg
tff(fact_8598_sum__nonneg,axiom,
    ! [A3: set_real,Fa: fun_real_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Fa,X2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(Fa,A3)) ) ).

% sum_nonneg
tff(fact_8599_sum__nonneg,axiom,
    ! [A3: set_nat,Fa: fun_nat_rat] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(Fa,A3)) ) ).

% sum_nonneg
tff(fact_8600_sum__nonneg,axiom,
    ! [A3: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_rat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(Fa,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups136491112297645522BT_rat(Fa,A3)) ) ).

% sum_nonneg
tff(fact_8601_sum__nonneg,axiom,
    ! [A3: set_int,Fa: fun_int_rat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(Fa,A3)) ) ).

% sum_nonneg
tff(fact_8602_sum__nonneg,axiom,
    ! [A3: set_real,Fa: fun_real_rat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(Fa,X2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(Fa,A3)) ) ).

% sum_nonneg
tff(fact_8603_sum__nonneg,axiom,
    ! [A3: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_VEBT_VEBT_nat(Fa,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups771621172384141258BT_nat(Fa,A3)) ) ).

% sum_nonneg
tff(fact_8604_sum__nonneg,axiom,
    ! [A3: set_int,Fa: fun_int_nat] :
      ( ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(Fa,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(Fa,A3)) ) ).

% sum_nonneg
tff(fact_8605_sum__nonneg,axiom,
    ! [A3: set_real,Fa: fun_real_nat] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(Fa,X2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(Fa,A3)) ) ).

% sum_nonneg
tff(fact_8606_sum__mono__inv,axiom,
    ! [Fa: fun_VEBT_VEBT_rat,I5: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,I: vEBT_VEBT] :
      ( ( groups136491112297645522BT_rat(Fa,I5) = groups136491112297645522BT_rat(G,I5) )
     => ( ! [I2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(Fa,I2)),aa_VEBT_VEBT_rat(G,I2)) )
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),I5)
         => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
           => ( aa_VEBT_VEBT_rat(Fa,I) = aa_VEBT_VEBT_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8607_sum__mono__inv,axiom,
    ! [Fa: fun_real_rat,I5: set_real,G: fun_real_rat,I: real] :
      ( ( groups1300246762558778688al_rat(Fa,I5) = groups1300246762558778688al_rat(G,I5) )
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(Fa,I2)),aa_real_rat(G,I2)) )
       => ( aa_set_real_o(member_real(I),I5)
         => ( aa_set_real_o(finite_finite_real,I5)
           => ( aa_real_rat(Fa,I) = aa_real_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8608_sum__mono__inv,axiom,
    ! [Fa: fun_nat_rat,I5: set_nat,G: fun_nat_rat,I: nat] :
      ( ( groups2906978787729119204at_rat(Fa,I5) = groups2906978787729119204at_rat(G,I5) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,I2)),aa_nat_rat(G,I2)) )
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( aa_set_nat_o(finite_finite_nat,I5)
           => ( aa_nat_rat(Fa,I) = aa_nat_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8609_sum__mono__inv,axiom,
    ! [Fa: fun_int_rat,I5: set_int,G: fun_int_rat,I: int] :
      ( ( groups3906332499630173760nt_rat(Fa,I5) = groups3906332499630173760nt_rat(G,I5) )
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,I2)),aa_int_rat(G,I2)) )
       => ( aa_set_int_o(member_int(I),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_rat(Fa,I) = aa_int_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8610_sum__mono__inv,axiom,
    ! [Fa: fun_complex_rat,I5: set_complex,G: fun_complex_rat,I: complex] :
      ( ( groups5058264527183730370ex_rat(Fa,I5) = groups5058264527183730370ex_rat(G,I5) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(Fa,I2)),aa_complex_rat(G,I2)) )
       => ( aa_set_complex_o(member_complex(I),I5)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I5)
           => ( aa_complex_rat(Fa,I) = aa_complex_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8611_sum__mono__inv,axiom,
    ! [Fa: fun_Code_integer_rat,I5: set_Code_integer,G: fun_Code_integer_rat,I: code_integer] :
      ( ( groups6602215022474089585er_rat(Fa,I5) = groups6602215022474089585er_rat(G,I5) )
     => ( ! [I2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(Fa,I2)),aa_Code_integer_rat(G,I2)) )
       => ( aa_set5643387332196575111eger_o(member_Code_integer(I),I5)
         => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
           => ( aa_Code_integer_rat(Fa,I) = aa_Code_integer_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8612_sum__mono__inv,axiom,
    ! [Fa: fun_VEBT_VEBT_nat,I5: set_VEBT_VEBT,G: fun_VEBT_VEBT_nat,I: vEBT_VEBT] :
      ( ( groups771621172384141258BT_nat(Fa,I5) = groups771621172384141258BT_nat(G,I5) )
     => ( ! [I2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(Fa,I2)),aa_VEBT_VEBT_nat(G,I2)) )
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),I5)
         => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
           => ( aa_VEBT_VEBT_nat(Fa,I) = aa_VEBT_VEBT_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8613_sum__mono__inv,axiom,
    ! [Fa: fun_real_nat,I5: set_real,G: fun_real_nat,I: real] :
      ( ( groups1935376822645274424al_nat(Fa,I5) = groups1935376822645274424al_nat(G,I5) )
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(Fa,I2)),aa_real_nat(G,I2)) )
       => ( aa_set_real_o(member_real(I),I5)
         => ( aa_set_real_o(finite_finite_real,I5)
           => ( aa_real_nat(Fa,I) = aa_real_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8614_sum__mono__inv,axiom,
    ! [Fa: fun_int_nat,I5: set_int,G: fun_int_nat,I: int] :
      ( ( groups4541462559716669496nt_nat(Fa,I5) = groups4541462559716669496nt_nat(G,I5) )
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Fa,I2)),aa_int_nat(G,I2)) )
       => ( aa_set_int_o(member_int(I),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_nat(Fa,I) = aa_int_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8615_sum__mono__inv,axiom,
    ! [Fa: fun_complex_nat,I5: set_complex,G: fun_complex_nat,I: complex] :
      ( ( groups5693394587270226106ex_nat(Fa,I5) = groups5693394587270226106ex_nat(G,I5) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(Fa,I2)),aa_complex_nat(G,I2)) )
       => ( aa_set_complex_o(member_complex(I),I5)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I5)
           => ( aa_complex_nat(Fa,I) = aa_complex_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_8616_sum__cong__Suc,axiom,
    ! [A3: set_nat,Fa: fun_nat_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A3)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X2)),A3)
           => ( aa_nat_nat(Fa,aa_nat_nat(suc,X2)) = aa_nat_nat(G,aa_nat_nat(suc,X2)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(Fa),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) ) ) ) ).

% sum_cong_Suc
tff(fact_8617_sum__cong__Suc,axiom,
    ! [A3: set_nat,Fa: fun_nat_real,G: fun_nat_real] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A3)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X2)),A3)
           => ( aa_nat_real(Fa,aa_nat_nat(suc,X2)) = aa_nat_real(G,aa_nat_nat(suc,X2)) ) )
       => ( groups6591440286371151544t_real(Fa,A3) = groups6591440286371151544t_real(G,A3) ) ) ) ).

% sum_cong_Suc
tff(fact_8618_sum_Ointer__filter,axiom,
    ! [A3: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,P: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
     => ( groups2240296850493347238T_real(G,collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_alq(A3),P))) = groups2240296850493347238T_real(aa_fun3597767889293385422T_real(aTP_Lamm_alr(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8619_sum_Ointer__filter,axiom,
    ! [A3: set_real,G: fun_real_real,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A3)
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_als(A3),P))) = groups8097168146408367636l_real(aa_fun4448911596306054750l_real(aTP_Lamm_alt(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8620_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_real,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_alu(A3),P))) = groups8778361861064173332t_real(aa_fun7506669883404309598t_real(aTP_Lamm_alv(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8621_sum_Ointer__filter,axiom,
    ! [A3: set_complex,G: fun_complex_real,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_alw(A3),P))) = groups5808333547571424918x_real(aa_fun8261057609414086622x_real(aTP_Lamm_alx(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8622_sum_Ointer__filter,axiom,
    ! [A3: set_Code_integer,G: fun_Co3960896333048504454r_real,P: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups1270011288395367621r_real(G,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_aly(A3),P))) = groups1270011288395367621r_real(aa_fun7534015886319449118r_real(aTP_Lamm_alz(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8623_sum_Ointer__filter,axiom,
    ! [A3: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,P: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
     => ( groups136491112297645522BT_rat(G,collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_alq(A3),P))) = groups136491112297645522BT_rat(aa_fun7833968636320851834BT_rat(aTP_Lamm_ama(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8624_sum_Ointer__filter,axiom,
    ! [A3: set_real,G: fun_real_rat,P: fun_real_o] :
      ( aa_set_real_o(finite_finite_real,A3)
     => ( groups1300246762558778688al_rat(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_als(A3),P))) = groups1300246762558778688al_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_amb(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8625_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_rat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( groups2906978787729119204at_rat(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_amc(A3),P))) = groups2906978787729119204at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_amd(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8626_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_rat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups3906332499630173760nt_rat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_alu(A3),P))) = groups3906332499630173760nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_ame(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8627_sum_Ointer__filter,axiom,
    ! [A3: set_complex,G: fun_complex_rat,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5058264527183730370ex_rat(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_alw(A3),P))) = groups5058264527183730370ex_rat(aa_fun4010016270681809034ex_rat(aTP_Lamm_amf(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_8628_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_amg(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_8629_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,M: nat,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N))) = groups6591440286371151544t_real(aTP_Lamm_vl(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_8630_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_amh(G),K)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_8631_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,M: nat,K: nat,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K))) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ur(G),K),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_8632_suminf__sum,axiom,
    ! [I5: set_VEBT_VEBT,Fa: fun_VE2989161857680785728t_real] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
         => summable_real(aa_VEB3736018289492012273t_real(Fa,I2)) )
     => ( suminf_real(aa_fun6147343916763992189t_real(aTP_Lamm_akz(I5),Fa)) = groups2240296850493347238T_real(aTP_Lamm_ami(Fa),I5) ) ) ).

% suminf_sum
tff(fact_8633_suminf__sum,axiom,
    ! [I5: set_int,Fa: fun_int_fun_nat_real] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I5)
         => summable_real(aa_int_fun_nat_real(Fa,I2)) )
     => ( suminf_real(aa_fun2842512154077127769t_real(aTP_Lamm_alb(I5),Fa)) = groups8778361861064173332t_real(aTP_Lamm_amj(Fa),I5) ) ) ).

% suminf_sum
tff(fact_8634_suminf__sum,axiom,
    ! [I5: set_real,Fa: fun_re4238438383274954274t_real] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),I5)
         => summable_real(aa_real_fun_nat_real(Fa,I2)) )
     => ( suminf_real(aa_fun628238740774635481t_real(aTP_Lamm_ald(I5),Fa)) = groups8097168146408367636l_real(aTP_Lamm_amk(Fa),I5) ) ) ).

% suminf_sum
tff(fact_8635_suminf__sum,axiom,
    ! [I5: set_nat,Fa: fun_na1142814060743658952omplex] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => summable_complex(aa_nat6454746962644518025omplex(Fa,I2)) )
     => ( suminf_complex(aa_fun4886358607025424513omplex(aTP_Lamm_alf(I5),Fa)) = groups2073611262835488442omplex(aTP_Lamm_aml(Fa),I5) ) ) ).

% suminf_sum
tff(fact_8636_suminf__sum,axiom,
    ! [I5: set_VEBT_VEBT,Fa: fun_VE6440932821983644738omplex] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
         => summable_complex(aa_VEB6050840761258413811omplex(Fa,I2)) )
     => ( suminf_complex(aa_fun3960763739589619457omplex(aTP_Lamm_alh(I5),Fa)) = groups1794756597179926696omplex(aTP_Lamm_amm(Fa),I5) ) ) ).

% suminf_sum
tff(fact_8637_suminf__sum,axiom,
    ! [I5: set_int,Fa: fun_in8056368165505483812omplex] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I5)
         => summable_complex(aa_int4144929030551567077omplex(Fa,I2)) )
     => ( suminf_complex(aa_fun3543561798013398749omplex(aTP_Lamm_alj(I5),Fa)) = groups3049146728041665814omplex(aTP_Lamm_amn(Fa),I5) ) ) ).

% suminf_sum
tff(fact_8638_suminf__sum,axiom,
    ! [I5: set_real,Fa: fun_re2084383175013312036omplex] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),I5)
         => summable_complex(aa_rea1531280491672885477omplex(Fa,I2)) )
     => ( suminf_complex(aa_fun7632768522510439005omplex(aTP_Lamm_all(I5),Fa)) = groups5754745047067104278omplex(aTP_Lamm_amo(Fa),I5) ) ) ).

% suminf_sum
tff(fact_8639_suminf__sum,axiom,
    ! [I5: set_nat,Fa: fun_nat_fun_nat_nat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => summable_nat(aa_nat_fun_nat_nat(Fa,I2)) )
     => ( suminf_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_alm(I5),Fa)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_amp(Fa)),I5) ) ) ).

% suminf_sum
tff(fact_8640_suminf__sum,axiom,
    ! [I5: set_complex,Fa: fun_co3258106519541737638omplex] :
      ( ! [I2: complex] :
          ( aa_set_complex_o(member_complex(I2),I5)
         => summable_complex(aa_com8181001469867462503omplex(Fa,I2)) )
     => ( suminf_complex(aa_fun5003373910569599327omplex(aTP_Lamm_alo(I5),Fa)) = groups7754918857620584856omplex(aTP_Lamm_amq(Fa),I5) ) ) ).

% suminf_sum
tff(fact_8641_suminf__sum,axiom,
    ! [I5: set_nat,Fa: fun_nat_fun_nat_real] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => summable_real(aa_nat_fun_nat_real(Fa,I2)) )
     => ( suminf_real(aa_fun7039663905532832253t_real(aTP_Lamm_alp(I5),Fa)) = groups6591440286371151544t_real(aTP_Lamm_amr(Fa),I5) ) ) ).

% suminf_sum
tff(fact_8642_summable__of__real,axiom,
    ! [X5: fun_nat_real] :
      ( summable_real(X5)
     => summable_real(aTP_Lamm_agv(X5)) ) ).

% summable_of_real
tff(fact_8643_summable__of__real,axiom,
    ! [X5: fun_nat_real] :
      ( summable_real(X5)
     => summable_complex(aTP_Lamm_agg(X5)) ) ).

% summable_of_real
tff(fact_8644_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(Fa,X2)) )
       => ( ( groups2240296850493347238T_real(Fa,A3) = zero_zero_real )
        <=> ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A3)
             => ( aa_VEBT_VEBT_real(Fa,X) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8645_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_real,Fa: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A3)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),A3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Fa,X2)) )
       => ( ( groups8097168146408367636l_real(Fa,A3) = zero_zero_real )
        <=> ! [X: real] :
              ( aa_set_real_o(member_real(X),A3)
             => ( aa_real_real(Fa,X) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8646_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,Fa: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Fa,X2)) )
       => ( ( groups8778361861064173332t_real(Fa,A3) = zero_zero_real )
        <=> ! [X: int] :
              ( aa_set_int_o(member_int(X),A3)
             => ( aa_int_real(Fa,X) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8647_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_complex,Fa: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(Fa,X2)) )
       => ( ( groups5808333547571424918x_real(Fa,A3) = zero_zero_real )
        <=> ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),A3)
             => ( aa_complex_real(Fa,X) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8648_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_Code_integer,Fa: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ! [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(Fa,X2)) )
       => ( ( groups1270011288395367621r_real(Fa,A3) = zero_zero_real )
        <=> ! [X: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X),A3)
             => ( aa_Code_integer_real(Fa,X) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8649_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(Fa,X2)) )
       => ( ( groups136491112297645522BT_rat(Fa,A3) = zero_zero_rat )
        <=> ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),A3)
             => ( aa_VEBT_VEBT_rat(Fa,X) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8650_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_real,Fa: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A3)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(Fa,X2)) )
       => ( ( groups1300246762558778688al_rat(Fa,A3) = zero_zero_rat )
        <=> ! [X: real] :
              ( aa_set_real_o(member_real(X),A3)
             => ( aa_real_rat(Fa,X) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8651_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,X2)) )
       => ( ( groups2906978787729119204at_rat(Fa,A3) = zero_zero_rat )
        <=> ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),A3)
             => ( aa_nat_rat(Fa,X) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8652_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,X2)) )
       => ( ( groups3906332499630173760nt_rat(Fa,A3) = zero_zero_rat )
        <=> ! [X: int] :
              ( aa_set_int_o(member_int(X),A3)
             => ( aa_int_rat(Fa,X) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8653_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_complex,Fa: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(Fa,X2)) )
       => ( ( groups5058264527183730370ex_rat(Fa,A3) = zero_zero_rat )
        <=> ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),A3)
             => ( aa_complex_rat(Fa,X) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_8654_sum__le__included,axiom,
    ! [S: set_int,Tb: set_int,G: fun_int_real,I: fun_int_int,Fa: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(finite_finite_int,Tb)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),Tb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X2)) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),Tb)
                    & ( aa_int_int(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(Fa,X2)),aa_int_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(Fa,S)),groups8778361861064173332t_real(G,Tb)) ) ) ) ) ).

% sum_le_included
tff(fact_8655_sum__le__included,axiom,
    ! [S: set_int,Tb: set_complex,G: fun_complex_real,I: fun_complex_int,Fa: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Tb)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),Tb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X2)) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Tb)
                    & ( aa_complex_int(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(Fa,X2)),aa_complex_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(Fa,S)),groups5808333547571424918x_real(G,Tb)) ) ) ) ) ).

% sum_le_included
tff(fact_8656_sum__le__included,axiom,
    ! [S: set_int,Tb: set_Code_integer,G: fun_Co3960896333048504454r_real,I: fun_Code_integer_int,Fa: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Tb)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),Tb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(G,X2)) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S)
               => ? [Xa: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),Tb)
                    & ( aa_Code_integer_int(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(Fa,X2)),aa_Code_integer_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(Fa,S)),groups1270011288395367621r_real(G,Tb)) ) ) ) ) ).

% sum_le_included
tff(fact_8657_sum__le__included,axiom,
    ! [S: set_complex,Tb: set_int,G: fun_int_real,I: fun_int_complex,Fa: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( aa_set_int_o(finite_finite_int,Tb)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),Tb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X2)) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),Tb)
                    & ( aa_int_complex(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(Fa,X2)),aa_int_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(Fa,S)),groups8778361861064173332t_real(G,Tb)) ) ) ) ) ).

% sum_le_included
tff(fact_8658_sum__le__included,axiom,
    ! [S: set_complex,Tb: set_complex,G: fun_complex_real,I: fun_complex_complex,Fa: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Tb)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),Tb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X2)) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Tb)
                    & ( aa_complex_complex(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(Fa,X2)),aa_complex_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(Fa,S)),groups5808333547571424918x_real(G,Tb)) ) ) ) ) ).

% sum_le_included
tff(fact_8659_sum__le__included,axiom,
    ! [S: set_complex,Tb: set_Code_integer,G: fun_Co3960896333048504454r_real,I: fun_Co1900716531650368264omplex,Fa: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Tb)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),Tb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(G,X2)) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S)
               => ? [Xa: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),Tb)
                    & ( aa_Cod5621350732091595593omplex(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(Fa,X2)),aa_Code_integer_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(Fa,S)),groups1270011288395367621r_real(G,Tb)) ) ) ) ) ).

% sum_le_included
tff(fact_8660_sum__le__included,axiom,
    ! [S: set_Code_integer,Tb: set_int,G: fun_int_real,I: fun_int_Code_integer,Fa: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set_int_o(finite_finite_int,Tb)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),Tb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X2)) )
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),Tb)
                    & ( aa_int_Code_integer(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(Fa,X2)),aa_int_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1270011288395367621r_real(Fa,S)),groups8778361861064173332t_real(G,Tb)) ) ) ) ) ).

% sum_le_included
tff(fact_8661_sum__le__included,axiom,
    ! [S: set_Code_integer,Tb: set_complex,G: fun_complex_real,I: fun_co8153928105989009288nteger,Fa: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Tb)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),Tb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X2)) )
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Tb)
                    & ( aa_com4217951221296830281nteger(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(Fa,X2)),aa_complex_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1270011288395367621r_real(Fa,S)),groups5808333547571424918x_real(G,Tb)) ) ) ) ) ).

% sum_le_included
tff(fact_8662_sum__le__included,axiom,
    ! [S: set_Code_integer,Tb: set_Code_integer,G: fun_Co3960896333048504454r_real,I: fun_Co8499238673658658103nteger,Fa: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,Tb)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),Tb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(G,X2)) )
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S)
               => ? [Xa: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),Tb)
                    & ( aa_Cod1783974652533396728nteger(I,Xa) = X2 )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(Fa,X2)),aa_Code_integer_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1270011288395367621r_real(Fa,S)),groups1270011288395367621r_real(G,Tb)) ) ) ) ) ).

% sum_le_included
tff(fact_8663_sum__le__included,axiom,
    ! [S: set_nat,Tb: set_nat,G: fun_nat_rat,I: fun_nat_nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(finite_finite_nat,Tb)
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),Tb)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X2)) )
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Tb)
                    & ( aa_nat_nat(I,Xa) = X2 )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,X2)),aa_nat_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(Fa,S)),groups2906978787729119204at_rat(G,Tb)) ) ) ) ) ).

% sum_le_included
tff(fact_8664_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,Fa: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(Fa,X2)),aa_int_real(G,X2)) )
       => ( ? [X3: int] :
              ( aa_set_int_o(member_int(X3),A3)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(Fa,X3)),aa_int_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(Fa,A3)),groups8778361861064173332t_real(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8665_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,Fa: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(Fa,X2)),aa_complex_real(G,X2)) )
       => ( ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A3)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(Fa,X3)),aa_complex_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(Fa,A3)),groups5808333547571424918x_real(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8666_sum__strict__mono__ex1,axiom,
    ! [A3: set_Code_integer,Fa: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ! [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(Fa,X2)),aa_Code_integer_real(G,X2)) )
       => ( ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Code_integer_real(Fa,X3)),aa_Code_integer_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups1270011288395367621r_real(Fa,A3)),groups1270011288395367621r_real(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8667_sum__strict__mono__ex1,axiom,
    ! [A3: set_nat,Fa: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,X2)),aa_nat_rat(G,X2)) )
       => ( ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A3)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,X3)),aa_nat_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(Fa,A3)),groups2906978787729119204at_rat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8668_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,Fa: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,X2)),aa_int_rat(G,X2)) )
       => ( ? [X3: int] :
              ( aa_set_int_o(member_int(X3),A3)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(Fa,X3)),aa_int_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(Fa,A3)),groups3906332499630173760nt_rat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8669_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,Fa: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(Fa,X2)),aa_complex_rat(G,X2)) )
       => ( ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A3)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(Fa,X3)),aa_complex_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(Fa,A3)),groups5058264527183730370ex_rat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8670_sum__strict__mono__ex1,axiom,
    ! [A3: set_Code_integer,Fa: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ! [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(Fa,X2)),aa_Code_integer_rat(G,X2)) )
       => ( ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Code_integer_rat(Fa,X3)),aa_Code_integer_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups6602215022474089585er_rat(Fa,A3)),groups6602215022474089585er_rat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8671_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,Fa: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Fa,X2)),aa_int_nat(G,X2)) )
       => ( ? [X3: int] :
              ( aa_set_int_o(member_int(X3),A3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(Fa,X3)),aa_int_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(Fa,A3)),groups4541462559716669496nt_nat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8672_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,Fa: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(Fa,X2)),aa_complex_nat(G,X2)) )
       => ( ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(Fa,X3)),aa_complex_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(Fa,A3)),groups5693394587270226106ex_nat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8673_sum__strict__mono__ex1,axiom,
    ! [A3: set_Code_integer,Fa: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ! [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(Fa,X2)),aa_Code_integer_nat(G,X2)) )
       => ( ? [X3: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(Fa,X3)),aa_Code_integer_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups7237345082560585321er_nat(Fa,A3)),groups7237345082560585321er_nat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_8674_sum_Orelated,axiom,
    ! [R4: fun_real_fun_real_o,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_real_o(aa_real_fun_real_o(R4,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R4,X15),X23)
              & aa_real_o(aa_real_fun_real_o(R4,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R4,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S3)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S3)
               => aa_real_o(aa_real_fun_real_o(R4,aa_int_real(H,X2)),aa_int_real(G,X2)) )
           => aa_real_o(aa_real_fun_real_o(R4,groups8778361861064173332t_real(H,S3)),groups8778361861064173332t_real(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_8675_sum_Orelated,axiom,
    ! [R4: fun_real_fun_real_o,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_real_o(aa_real_fun_real_o(R4,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R4,X15),X23)
              & aa_real_o(aa_real_fun_real_o(R4,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R4,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S3)
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => aa_real_o(aa_real_fun_real_o(R4,aa_complex_real(H,X2)),aa_complex_real(G,X2)) )
           => aa_real_o(aa_real_fun_real_o(R4,groups5808333547571424918x_real(H,S3)),groups5808333547571424918x_real(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_8676_sum_Orelated,axiom,
    ! [R4: fun_real_fun_real_o,S3: set_Code_integer,H: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( aa_real_o(aa_real_fun_real_o(R4,zero_zero_real),zero_zero_real)
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R4,X15),X23)
              & aa_real_o(aa_real_fun_real_o(R4,Y15),Y23) )
           => aa_real_o(aa_real_fun_real_o(R4,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X15),Y15)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S3)
               => aa_real_o(aa_real_fun_real_o(R4,aa_Code_integer_real(H,X2)),aa_Code_integer_real(G,X2)) )
           => aa_real_o(aa_real_fun_real_o(R4,groups1270011288395367621r_real(H,S3)),groups1270011288395367621r_real(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_8677_sum_Orelated,axiom,
    ! [R4: fun_rat_fun_rat_o,S3: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R4,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R4,X15),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R4,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R4,plus_plus_rat(X15,Y15)),plus_plus_rat(X23,Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S3)
         => ( ! [X2: nat] :
                ( aa_set_nat_o(member_nat(X2),S3)
               => aa_rat_o(aa_rat_fun_rat_o(R4,aa_nat_rat(H,X2)),aa_nat_rat(G,X2)) )
           => aa_rat_o(aa_rat_fun_rat_o(R4,groups2906978787729119204at_rat(H,S3)),groups2906978787729119204at_rat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_8678_sum_Orelated,axiom,
    ! [R4: fun_rat_fun_rat_o,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R4,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R4,X15),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R4,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R4,plus_plus_rat(X15,Y15)),plus_plus_rat(X23,Y23)) )
       => ( aa_set_int_o(finite_finite_int,S3)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S3)
               => aa_rat_o(aa_rat_fun_rat_o(R4,aa_int_rat(H,X2)),aa_int_rat(G,X2)) )
           => aa_rat_o(aa_rat_fun_rat_o(R4,groups3906332499630173760nt_rat(H,S3)),groups3906332499630173760nt_rat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_8679_sum_Orelated,axiom,
    ! [R4: fun_rat_fun_rat_o,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R4,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R4,X15),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R4,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R4,plus_plus_rat(X15,Y15)),plus_plus_rat(X23,Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S3)
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => aa_rat_o(aa_rat_fun_rat_o(R4,aa_complex_rat(H,X2)),aa_complex_rat(G,X2)) )
           => aa_rat_o(aa_rat_fun_rat_o(R4,groups5058264527183730370ex_rat(H,S3)),groups5058264527183730370ex_rat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_8680_sum_Orelated,axiom,
    ! [R4: fun_rat_fun_rat_o,S3: set_Code_integer,H: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R4,zero_zero_rat),zero_zero_rat)
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R4,X15),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R4,Y15),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R4,plus_plus_rat(X15,Y15)),plus_plus_rat(X23,Y23)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S3)
               => aa_rat_o(aa_rat_fun_rat_o(R4,aa_Code_integer_rat(H,X2)),aa_Code_integer_rat(G,X2)) )
           => aa_rat_o(aa_rat_fun_rat_o(R4,groups6602215022474089585er_rat(H,S3)),groups6602215022474089585er_rat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_8681_sum_Orelated,axiom,
    ! [R4: fun_nat_fun_nat_o,S3: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R4,zero_zero_nat),zero_zero_nat)
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R4,X15),X23)
              & aa_nat_o(aa_nat_fun_nat_o(R4,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R4,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S3)
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S3)
               => aa_nat_o(aa_nat_fun_nat_o(R4,aa_int_nat(H,X2)),aa_int_nat(G,X2)) )
           => aa_nat_o(aa_nat_fun_nat_o(R4,groups4541462559716669496nt_nat(H,S3)),groups4541462559716669496nt_nat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_8682_sum_Orelated,axiom,
    ! [R4: fun_nat_fun_nat_o,S3: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R4,zero_zero_nat),zero_zero_nat)
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R4,X15),X23)
              & aa_nat_o(aa_nat_fun_nat_o(R4,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R4,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S3)
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => aa_nat_o(aa_nat_fun_nat_o(R4,aa_complex_nat(H,X2)),aa_complex_nat(G,X2)) )
           => aa_nat_o(aa_nat_fun_nat_o(R4,groups5693394587270226106ex_nat(H,S3)),groups5693394587270226106ex_nat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_8683_sum_Orelated,axiom,
    ! [R4: fun_nat_fun_nat_o,S3: set_Code_integer,H: fun_Code_integer_nat,G: fun_Code_integer_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R4,zero_zero_nat),zero_zero_nat)
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R4,X15),X23)
              & aa_nat_o(aa_nat_fun_nat_o(R4,Y15),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R4,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X15),Y15)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S3)
               => aa_nat_o(aa_nat_fun_nat_o(R4,aa_Code_integer_nat(H,X2)),aa_Code_integer_nat(G,X2)) )
           => aa_nat_o(aa_nat_fun_nat_o(R4,groups7237345082560585321er_nat(H,S3)),groups7237345082560585321er_nat(G,S3)) ) ) ) ) ).

% sum.related
tff(fact_8684_sum__strict__mono,axiom,
    ! [A3: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
     => ( ( A3 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_VEBT_VEBT_real(Fa,X2)),aa_VEBT_VEBT_real(G,X2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups2240296850493347238T_real(Fa,A3)),groups2240296850493347238T_real(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_8685_sum__strict__mono,axiom,
    ! [A3: set_complex,Fa: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ( A3 != bot_bot_set_complex )
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),A3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(Fa,X2)),aa_complex_real(G,X2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(Fa,A3)),groups5808333547571424918x_real(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_8686_sum__strict__mono,axiom,
    ! [A3: set_Code_integer,Fa: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_Code_integer_real(Fa,X2)),aa_Code_integer_real(G,X2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups1270011288395367621r_real(Fa,A3)),groups1270011288395367621r_real(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_8687_sum__strict__mono,axiom,
    ! [A3: set_real,Fa: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,A3)
     => ( ( A3 != bot_bot_set_real )
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(Fa,X2)),aa_real_real(G,X2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(Fa,A3)),groups8097168146408367636l_real(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_8688_sum__strict__mono,axiom,
    ! [A3: set_int,Fa: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( A3 != bot_bot_set_int )
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),A3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(Fa,X2)),aa_int_real(G,X2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(Fa,A3)),groups8778361861064173332t_real(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_8689_sum__strict__mono,axiom,
    ! [A3: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
     => ( ( A3 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_VEBT_VEBT_rat(Fa,X2)),aa_VEBT_VEBT_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups136491112297645522BT_rat(Fa,A3)),groups136491112297645522BT_rat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_8690_sum__strict__mono,axiom,
    ! [A3: set_complex,Fa: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ( A3 != bot_bot_set_complex )
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),A3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(Fa,X2)),aa_complex_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(Fa,A3)),groups5058264527183730370ex_rat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_8691_sum__strict__mono,axiom,
    ! [A3: set_Code_integer,Fa: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( ( A3 != bot_bo3990330152332043303nteger )
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),A3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_Code_integer_rat(Fa,X2)),aa_Code_integer_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups6602215022474089585er_rat(Fa,A3)),groups6602215022474089585er_rat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_8692_sum__strict__mono,axiom,
    ! [A3: set_real,Fa: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,A3)
     => ( ( A3 != bot_bot_set_real )
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),A3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(Fa,X2)),aa_real_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(Fa,A3)),groups1300246762558778688al_rat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_8693_sum__strict__mono,axiom,
    ! [A3: set_nat,Fa: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [X2: nat] :
              ( aa_set_nat_o(member_nat(X2),A3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(Fa,X2)),aa_nat_rat(G,X2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(Fa,A3)),groups2906978787729119204at_rat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_8694_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T4: set_VEBT_VEBT,S3: set_VEBT_VEBT,I: fun_VE3165209603783583043T_VEBT,J: fun_VE3165209603783583043T_VEBT,T5: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S6)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T4)
       => ( ! [A: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(S3,S6))
             => ( aa_VEB5832953974749077948T_VEBT(I,aa_VEB5832953974749077948T_VEBT(J,A)) = A ) )
         => ( ! [A: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(S3,S6))
               => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(J,A)),minus_5127226145743854075T_VEBT(T5,T4)) )
           => ( ! [B: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(T5,T4))
                 => ( aa_VEB5832953974749077948T_VEBT(J,aa_VEB5832953974749077948T_VEBT(I,B)) = B ) )
             => ( ! [B: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(T5,T4))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_VEB5832953974749077948T_VEBT(I,B)),minus_5127226145743854075T_VEBT(S3,S6)) )
               => ( ! [A: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S6)
                     => ( aa_VEBT_VEBT_real(G,A) = zero_zero_real ) )
                 => ( ! [B: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),T4)
                       => ( aa_VEBT_VEBT_real(H,B) = zero_zero_real ) )
                   => ( ! [A: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3)
                         => ( aa_VEBT_VEBT_real(H,aa_VEB5832953974749077948T_VEBT(J,A)) = aa_VEBT_VEBT_real(G,A) ) )
                     => ( groups2240296850493347238T_real(G,S3) = groups2240296850493347238T_real(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8695_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T4: set_real,S3: set_VEBT_VEBT,I: fun_real_VEBT_VEBT,J: fun_VEBT_VEBT_real,T5: set_real,G: fun_VEBT_VEBT_real,H: fun_real_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S6)
     => ( aa_set_real_o(finite_finite_real,T4)
       => ( ! [A: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(S3,S6))
             => ( aa_real_VEBT_VEBT(I,aa_VEBT_VEBT_real(J,A)) = A ) )
         => ( ! [A: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(S3,S6))
               => aa_set_real_o(member_real(aa_VEBT_VEBT_real(J,A)),minus_minus_set_real(T5,T4)) )
           => ( ! [B: real] :
                  ( aa_set_real_o(member_real(B),minus_minus_set_real(T5,T4))
                 => ( aa_VEBT_VEBT_real(J,aa_real_VEBT_VEBT(I,B)) = B ) )
             => ( ! [B: real] :
                    ( aa_set_real_o(member_real(B),minus_minus_set_real(T5,T4))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_real_VEBT_VEBT(I,B)),minus_5127226145743854075T_VEBT(S3,S6)) )
               => ( ! [A: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S6)
                     => ( aa_VEBT_VEBT_real(G,A) = zero_zero_real ) )
                 => ( ! [B: real] :
                        ( aa_set_real_o(member_real(B),T4)
                       => ( aa_real_real(H,B) = zero_zero_real ) )
                   => ( ! [A: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3)
                         => ( aa_real_real(H,aa_VEBT_VEBT_real(J,A)) = aa_VEBT_VEBT_real(G,A) ) )
                     => ( groups2240296850493347238T_real(G,S3) = groups8097168146408367636l_real(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8696_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T4: set_VEBT_VEBT,S3: set_real,I: fun_VEBT_VEBT_real,J: fun_real_VEBT_VEBT,T5: set_VEBT_VEBT,G: fun_real_real,H: fun_VEBT_VEBT_real] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T4)
       => ( ! [A: real] :
              ( aa_set_real_o(member_real(A),minus_minus_set_real(S3,S6))
             => ( aa_VEBT_VEBT_real(I,aa_real_VEBT_VEBT(J,A)) = A ) )
         => ( ! [A: real] :
                ( aa_set_real_o(member_real(A),minus_minus_set_real(S3,S6))
               => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_real_VEBT_VEBT(J,A)),minus_5127226145743854075T_VEBT(T5,T4)) )
           => ( ! [B: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(T5,T4))
                 => ( aa_real_VEBT_VEBT(J,aa_VEBT_VEBT_real(I,B)) = B ) )
             => ( ! [B: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(T5,T4))
                   => aa_set_real_o(member_real(aa_VEBT_VEBT_real(I,B)),minus_minus_set_real(S3,S6)) )
               => ( ! [A: real] :
                      ( aa_set_real_o(member_real(A),S6)
                     => ( aa_real_real(G,A) = zero_zero_real ) )
                 => ( ! [B: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),T4)
                       => ( aa_VEBT_VEBT_real(H,B) = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( aa_set_real_o(member_real(A),S3)
                         => ( aa_VEBT_VEBT_real(H,aa_real_VEBT_VEBT(J,A)) = aa_real_real(G,A) ) )
                     => ( groups8097168146408367636l_real(G,S3) = groups2240296850493347238T_real(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8697_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T4: set_real,S3: set_real,I: fun_real_real,J: fun_real_real,T5: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_real_o(finite_finite_real,T4)
       => ( ! [A: real] :
              ( aa_set_real_o(member_real(A),minus_minus_set_real(S3,S6))
             => ( aa_real_real(I,aa_real_real(J,A)) = A ) )
         => ( ! [A: real] :
                ( aa_set_real_o(member_real(A),minus_minus_set_real(S3,S6))
               => aa_set_real_o(member_real(aa_real_real(J,A)),minus_minus_set_real(T5,T4)) )
           => ( ! [B: real] :
                  ( aa_set_real_o(member_real(B),minus_minus_set_real(T5,T4))
                 => ( aa_real_real(J,aa_real_real(I,B)) = B ) )
             => ( ! [B: real] :
                    ( aa_set_real_o(member_real(B),minus_minus_set_real(T5,T4))
                   => aa_set_real_o(member_real(aa_real_real(I,B)),minus_minus_set_real(S3,S6)) )
               => ( ! [A: real] :
                      ( aa_set_real_o(member_real(A),S6)
                     => ( aa_real_real(G,A) = zero_zero_real ) )
                 => ( ! [B: real] :
                        ( aa_set_real_o(member_real(B),T4)
                       => ( aa_real_real(H,B) = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( aa_set_real_o(member_real(A),S3)
                         => ( aa_real_real(H,aa_real_real(J,A)) = aa_real_real(G,A) ) )
                     => ( groups8097168146408367636l_real(G,S3) = groups8097168146408367636l_real(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8698_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T4: set_int,S3: set_VEBT_VEBT,I: fun_int_VEBT_VEBT,J: fun_VEBT_VEBT_int,T5: set_int,G: fun_VEBT_VEBT_real,H: fun_int_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S6)
     => ( aa_set_int_o(finite_finite_int,T4)
       => ( ! [A: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(S3,S6))
             => ( aa_int_VEBT_VEBT(I,aa_VEBT_VEBT_int(J,A)) = A ) )
         => ( ! [A: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(S3,S6))
               => aa_set_int_o(member_int(aa_VEBT_VEBT_int(J,A)),minus_minus_set_int(T5,T4)) )
           => ( ! [B: int] :
                  ( aa_set_int_o(member_int(B),minus_minus_set_int(T5,T4))
                 => ( aa_VEBT_VEBT_int(J,aa_int_VEBT_VEBT(I,B)) = B ) )
             => ( ! [B: int] :
                    ( aa_set_int_o(member_int(B),minus_minus_set_int(T5,T4))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_int_VEBT_VEBT(I,B)),minus_5127226145743854075T_VEBT(S3,S6)) )
               => ( ! [A: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S6)
                     => ( aa_VEBT_VEBT_real(G,A) = zero_zero_real ) )
                 => ( ! [B: int] :
                        ( aa_set_int_o(member_int(B),T4)
                       => ( aa_int_real(H,B) = zero_zero_real ) )
                   => ( ! [A: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3)
                         => ( aa_int_real(H,aa_VEBT_VEBT_int(J,A)) = aa_VEBT_VEBT_real(G,A) ) )
                     => ( groups2240296850493347238T_real(G,S3) = groups8778361861064173332t_real(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8699_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T4: set_int,S3: set_real,I: fun_int_real,J: fun_real_int,T5: set_int,G: fun_real_real,H: fun_int_real] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_int_o(finite_finite_int,T4)
       => ( ! [A: real] :
              ( aa_set_real_o(member_real(A),minus_minus_set_real(S3,S6))
             => ( aa_int_real(I,aa_real_int(J,A)) = A ) )
         => ( ! [A: real] :
                ( aa_set_real_o(member_real(A),minus_minus_set_real(S3,S6))
               => aa_set_int_o(member_int(aa_real_int(J,A)),minus_minus_set_int(T5,T4)) )
           => ( ! [B: int] :
                  ( aa_set_int_o(member_int(B),minus_minus_set_int(T5,T4))
                 => ( aa_real_int(J,aa_int_real(I,B)) = B ) )
             => ( ! [B: int] :
                    ( aa_set_int_o(member_int(B),minus_minus_set_int(T5,T4))
                   => aa_set_real_o(member_real(aa_int_real(I,B)),minus_minus_set_real(S3,S6)) )
               => ( ! [A: real] :
                      ( aa_set_real_o(member_real(A),S6)
                     => ( aa_real_real(G,A) = zero_zero_real ) )
                 => ( ! [B: int] :
                        ( aa_set_int_o(member_int(B),T4)
                       => ( aa_int_real(H,B) = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( aa_set_real_o(member_real(A),S3)
                         => ( aa_int_real(H,aa_real_int(J,A)) = aa_real_real(G,A) ) )
                     => ( groups8097168146408367636l_real(G,S3) = groups8778361861064173332t_real(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8700_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T4: set_complex,S3: set_VEBT_VEBT,I: fun_co4340422964966740447T_VEBT,J: fun_VE5569070535936029429omplex,T5: set_complex,G: fun_VEBT_VEBT_real,H: fun_complex_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S6)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T4)
       => ( ! [A: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(S3,S6))
             => ( aa_complex_VEBT_VEBT(I,aa_VEBT_VEBT_complex(J,A)) = A ) )
         => ( ! [A: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(S3,S6))
               => aa_set_complex_o(member_complex(aa_VEBT_VEBT_complex(J,A)),minus_811609699411566653omplex(T5,T4)) )
           => ( ! [B: complex] :
                  ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(T5,T4))
                 => ( aa_VEBT_VEBT_complex(J,aa_complex_VEBT_VEBT(I,B)) = B ) )
             => ( ! [B: complex] :
                    ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(T5,T4))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_complex_VEBT_VEBT(I,B)),minus_5127226145743854075T_VEBT(S3,S6)) )
               => ( ! [A: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S6)
                     => ( aa_VEBT_VEBT_real(G,A) = zero_zero_real ) )
                 => ( ! [B: complex] :
                        ( aa_set_complex_o(member_complex(B),T4)
                       => ( aa_complex_real(H,B) = zero_zero_real ) )
                   => ( ! [A: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3)
                         => ( aa_complex_real(H,aa_VEBT_VEBT_complex(J,A)) = aa_VEBT_VEBT_real(G,A) ) )
                     => ( groups2240296850493347238T_real(G,S3) = groups5808333547571424918x_real(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8701_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T4: set_complex,S3: set_real,I: fun_complex_real,J: fun_real_complex,T5: set_complex,G: fun_real_real,H: fun_complex_real] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T4)
       => ( ! [A: real] :
              ( aa_set_real_o(member_real(A),minus_minus_set_real(S3,S6))
             => ( aa_complex_real(I,aa_real_complex(J,A)) = A ) )
         => ( ! [A: real] :
                ( aa_set_real_o(member_real(A),minus_minus_set_real(S3,S6))
               => aa_set_complex_o(member_complex(aa_real_complex(J,A)),minus_811609699411566653omplex(T5,T4)) )
           => ( ! [B: complex] :
                  ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(T5,T4))
                 => ( aa_real_complex(J,aa_complex_real(I,B)) = B ) )
             => ( ! [B: complex] :
                    ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(T5,T4))
                   => aa_set_real_o(member_real(aa_complex_real(I,B)),minus_minus_set_real(S3,S6)) )
               => ( ! [A: real] :
                      ( aa_set_real_o(member_real(A),S6)
                     => ( aa_real_real(G,A) = zero_zero_real ) )
                 => ( ! [B: complex] :
                        ( aa_set_complex_o(member_complex(B),T4)
                       => ( aa_complex_real(H,B) = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( aa_set_real_o(member_real(A),S3)
                         => ( aa_complex_real(H,aa_real_complex(J,A)) = aa_real_real(G,A) ) )
                     => ( groups8097168146408367636l_real(G,S3) = groups5808333547571424918x_real(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8702_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_VEBT_VEBT,T4: set_Code_integer,S3: set_VEBT_VEBT,I: fun_Co3766629010929971184T_VEBT,J: fun_VE9219535225822561060nteger,T5: set_Code_integer,G: fun_VEBT_VEBT_real,H: fun_Co3960896333048504454r_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S6)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
       => ( ! [A: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(S3,S6))
             => ( aa_Cod6507154402108480601T_VEBT(I,aa_VEB5935388676523804373nteger(J,A)) = A ) )
         => ( ! [A: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(S3,S6))
               => aa_set5643387332196575111eger_o(member_Code_integer(aa_VEB5935388676523804373nteger(J,A)),minus_2355218937544613996nteger(T5,T4)) )
           => ( ! [B: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(T5,T4))
                 => ( aa_VEB5935388676523804373nteger(J,aa_Cod6507154402108480601T_VEBT(I,B)) = B ) )
             => ( ! [B: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(T5,T4))
                   => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_Cod6507154402108480601T_VEBT(I,B)),minus_5127226145743854075T_VEBT(S3,S6)) )
               => ( ! [A: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S6)
                     => ( aa_VEBT_VEBT_real(G,A) = zero_zero_real ) )
                 => ( ! [B: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(B),T4)
                       => ( aa_Code_integer_real(H,B) = zero_zero_real ) )
                   => ( ! [A: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),S3)
                         => ( aa_Code_integer_real(H,aa_VEB5935388676523804373nteger(J,A)) = aa_VEBT_VEBT_real(G,A) ) )
                     => ( groups2240296850493347238T_real(G,S3) = groups1270011288395367621r_real(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8703_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T4: set_Code_integer,S3: set_real,I: fun_Co3960896333048504454r_real,J: fun_re6978948838062957830nteger,T5: set_Code_integer,G: fun_real_real,H: fun_Co3960896333048504454r_real] :
      ( aa_set_real_o(finite_finite_real,S6)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T4)
       => ( ! [A: real] :
              ( aa_set_real_o(member_real(A),minus_minus_set_real(S3,S6))
             => ( aa_Code_integer_real(I,aa_real_Code_integer(J,A)) = A ) )
         => ( ! [A: real] :
                ( aa_set_real_o(member_real(A),minus_minus_set_real(S3,S6))
               => aa_set5643387332196575111eger_o(member_Code_integer(aa_real_Code_integer(J,A)),minus_2355218937544613996nteger(T5,T4)) )
           => ( ! [B: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(T5,T4))
                 => ( aa_real_Code_integer(J,aa_Code_integer_real(I,B)) = B ) )
             => ( ! [B: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(T5,T4))
                   => aa_set_real_o(member_real(aa_Code_integer_real(I,B)),minus_minus_set_real(S3,S6)) )
               => ( ! [A: real] :
                      ( aa_set_real_o(member_real(A),S6)
                     => ( aa_real_real(G,A) = zero_zero_real ) )
                 => ( ! [B: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(B),T4)
                       => ( aa_Code_integer_real(H,B) = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( aa_set_real_o(member_real(A),S3)
                         => ( aa_Code_integer_real(H,aa_real_Code_integer(J,A)) = aa_real_real(G,A) ) )
                     => ( groups8097168146408367636l_real(G,S3) = groups1270011288395367621r_real(H,T5) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_8704_nonzero__of__real__divide,axiom,
    ! [Y: real,Xb: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(Xb),Y)) = aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y)) ) ) ).

% nonzero_of_real_divide
tff(fact_8705_nonzero__of__real__divide,axiom,
    ! [Y: real,Xb: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(divide_divide_real(Xb),Y)) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y)) ) ) ).

% nonzero_of_real_divide
tff(fact_8706_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_real,B5: real,I: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S)
     => ( ! [I2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(Fa,I2)) )
       => ( ( groups2240296850493347238T_real(Fa,S) = B5 )
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(Fa,I)),B5) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8707_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,Fa: fun_real_real,B5: real,I: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Fa,I2)) )
       => ( ( groups8097168146408367636l_real(Fa,S) = B5 )
         => ( aa_set_real_o(member_real(I),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(Fa,I)),B5) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8708_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,Fa: fun_int_real,B5: real,I: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Fa,I2)) )
       => ( ( groups8778361861064173332t_real(Fa,S) = B5 )
         => ( aa_set_int_o(member_int(I),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(Fa,I)),B5) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8709_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,Fa: fun_complex_real,B5: real,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(Fa,I2)) )
       => ( ( groups5808333547571424918x_real(Fa,S) = B5 )
         => ( aa_set_complex_o(member_complex(I),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(Fa,I)),B5) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8710_sum__nonneg__leq__bound,axiom,
    ! [S: set_Code_integer,Fa: fun_Co3960896333048504454r_real,B5: real,I: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( ! [I2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(Fa,I2)) )
       => ( ( groups1270011288395367621r_real(Fa,S) = B5 )
         => ( aa_set5643387332196575111eger_o(member_Code_integer(I),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(Fa,I)),B5) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8711_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_rat,B5: rat,I: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S)
     => ( ! [I2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(Fa,I2)) )
       => ( ( groups136491112297645522BT_rat(Fa,S) = B5 )
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(Fa,I)),B5) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8712_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,Fa: fun_real_rat,B5: rat,I: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(Fa,I2)) )
       => ( ( groups1300246762558778688al_rat(Fa,S) = B5 )
         => ( aa_set_real_o(member_real(I),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(Fa,I)),B5) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8713_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,Fa: fun_nat_rat,B5: rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,I2)) )
       => ( ( groups2906978787729119204at_rat(Fa,S) = B5 )
         => ( aa_set_nat_o(member_nat(I),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(Fa,I)),B5) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8714_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,Fa: fun_int_rat,B5: rat,I: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,I2)) )
       => ( ( groups3906332499630173760nt_rat(Fa,S) = B5 )
         => ( aa_set_int_o(member_int(I),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,I)),B5) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8715_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,Fa: fun_complex_rat,B5: rat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(Fa,I2)) )
       => ( ( groups5058264527183730370ex_rat(Fa,S) = B5 )
         => ( aa_set_complex_o(member_complex(I),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(Fa,I)),B5) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_8716_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_real,I: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S)
     => ( ! [I2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(Fa,I2)) )
       => ( ( groups2240296850493347238T_real(Fa,S) = zero_zero_real )
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),S)
           => ( aa_VEBT_VEBT_real(Fa,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8717_sum__nonneg__0,axiom,
    ! [S: set_real,Fa: fun_real_real,I: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Fa,I2)) )
       => ( ( groups8097168146408367636l_real(Fa,S) = zero_zero_real )
         => ( aa_set_real_o(member_real(I),S)
           => ( aa_real_real(Fa,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8718_sum__nonneg__0,axiom,
    ! [S: set_int,Fa: fun_int_real,I: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Fa,I2)) )
       => ( ( groups8778361861064173332t_real(Fa,S) = zero_zero_real )
         => ( aa_set_int_o(member_int(I),S)
           => ( aa_int_real(Fa,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8719_sum__nonneg__0,axiom,
    ! [S: set_complex,Fa: fun_complex_real,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(Fa,I2)) )
       => ( ( groups5808333547571424918x_real(Fa,S) = zero_zero_real )
         => ( aa_set_complex_o(member_complex(I),S)
           => ( aa_complex_real(Fa,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8720_sum__nonneg__0,axiom,
    ! [S: set_Code_integer,Fa: fun_Co3960896333048504454r_real,I: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( ! [I2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(I2),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(Fa,I2)) )
       => ( ( groups1270011288395367621r_real(Fa,S) = zero_zero_real )
         => ( aa_set5643387332196575111eger_o(member_Code_integer(I),S)
           => ( aa_Code_integer_real(Fa,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8721_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_rat,I: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S)
     => ( ! [I2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(Fa,I2)) )
       => ( ( groups136491112297645522BT_rat(Fa,S) = zero_zero_rat )
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),S)
           => ( aa_VEBT_VEBT_rat(Fa,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8722_sum__nonneg__0,axiom,
    ! [S: set_real,Fa: fun_real_rat,I: real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ! [I2: real] :
            ( aa_set_real_o(member_real(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(Fa,I2)) )
       => ( ( groups1300246762558778688al_rat(Fa,S) = zero_zero_rat )
         => ( aa_set_real_o(member_real(I),S)
           => ( aa_real_rat(Fa,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8723_sum__nonneg__0,axiom,
    ! [S: set_nat,Fa: fun_nat_rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,I2)) )
       => ( ( groups2906978787729119204at_rat(Fa,S) = zero_zero_rat )
         => ( aa_set_nat_o(member_nat(I),S)
           => ( aa_nat_rat(Fa,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8724_sum__nonneg__0,axiom,
    ! [S: set_int,Fa: fun_int_rat,I: int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,I2)) )
       => ( ( groups3906332499630173760nt_rat(Fa,S) = zero_zero_rat )
         => ( aa_set_int_o(member_int(I),S)
           => ( aa_int_rat(Fa,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8725_sum__nonneg__0,axiom,
    ! [S: set_complex,Fa: fun_complex_rat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(Fa,I2)) )
       => ( ( groups5058264527183730370ex_rat(Fa,S) = zero_zero_rat )
         => ( aa_set_complex_o(member_complex(I),S)
           => ( aa_complex_rat(Fa,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_8726_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_ams(G)))) = groups8778361861064173332t_real(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8727_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_amt(G)))) = groups5808333547571424918x_real(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8728_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups1270011288395367621r_real(G,minus_2355218937544613996nteger(A3,collect_Code_integer(aTP_Lamm_amu(G)))) = groups1270011288395367621r_real(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8729_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_amv(G)))) = groups3906332499630173760nt_rat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8730_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_amw(G)))) = groups5058264527183730370ex_rat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8731_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups6602215022474089585er_rat(G,minus_2355218937544613996nteger(A3,collect_Code_integer(aTP_Lamm_amx(G)))) = groups6602215022474089585er_rat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8732_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_amy(G)))) = groups4541462559716669496nt_nat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8733_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_amz(G)))) = groups5693394587270226106ex_nat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8734_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
     => ( groups7237345082560585321er_nat(G,minus_2355218937544613996nteger(A3,collect_Code_integer(aTP_Lamm_ana(G)))) = groups7237345082560585321er_nat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8735_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_anb(G)))) = groups5690904116761175830ex_int(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_8736_sum__power__add,axiom,
    ! [Xb: complex,M: nat,I5: set_nat] : groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_anc(Xb),M),I5) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Xb),M)),groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,Xb),I5)) ).

% sum_power_add
tff(fact_8737_sum__power__add,axiom,
    ! [Xb: code_integer,M: nat,I5: set_nat] : groups7501900531339628137nteger(aa_nat2010205248571845816nteger(aTP_Lamm_and(Xb),M),I5) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),M)),groups7501900531339628137nteger(power_8256067586552552935nteger(Xb),I5)) ).

% sum_power_add
tff(fact_8738_sum__power__add,axiom,
    ! [Xb: rat,M: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_ane(Xb),M),I5) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),M)),groups2906978787729119204at_rat(power_power_rat(Xb),I5)) ).

% sum_power_add
tff(fact_8739_sum__power__add,axiom,
    ! [Xb: int,M: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_anf(Xb),M),I5) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),M)),groups3539618377306564664at_int(power_power_int(Xb),I5)) ).

% sum_power_add
tff(fact_8740_sum__power__add,axiom,
    ! [Xb: real,M: nat,I5: set_nat] : groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ang(Xb),M),I5) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),M)),groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb),I5)) ).

% sum_power_add
tff(fact_8741_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N),M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_anh(G),N),M)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N),M)) ).

% sum.atLeastAtMost_rev
tff(fact_8742_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,N: nat,M: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N),M)) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_ani(G),N),M),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,N),M)) ).

% sum.atLeastAtMost_rev
tff(fact_8743_suminf__finite,axiom,
    ! [N2: set_nat,Fa: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,N2)
     => ( ! [N3: nat] :
            ( ~ aa_set_nat_o(member_nat(N3),N2)
           => ( aa_nat_int(Fa,N3) = zero_zero_int ) )
       => ( suminf_int(Fa) = groups3539618377306564664at_int(Fa,N2) ) ) ) ).

% suminf_finite
tff(fact_8744_suminf__finite,axiom,
    ! [N2: set_nat,Fa: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,N2)
     => ( ! [N3: nat] :
            ( ~ aa_set_nat_o(member_nat(N3),N2)
           => ( aa_nat_nat(Fa,N3) = zero_zero_nat ) )
       => ( suminf_nat(Fa) = aa_set_nat_nat(groups3542108847815614940at_nat(Fa),N2) ) ) ) ).

% suminf_finite
tff(fact_8745_suminf__finite,axiom,
    ! [N2: set_nat,Fa: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,N2)
     => ( ! [N3: nat] :
            ( ~ aa_set_nat_o(member_nat(N3),N2)
           => ( aa_nat_real(Fa,N3) = zero_zero_real ) )
       => ( suminf_real(Fa) = groups6591440286371151544t_real(Fa,N2) ) ) ) ).

% suminf_finite
tff(fact_8746_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,Fa: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(Fa,I))
         => ( ! [I2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(Fa,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups2240296850493347238T_real(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8747_sum__pos2,axiom,
    ! [I5: set_real,I: real,Fa: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(Fa,I))
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Fa,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8748_sum__pos2,axiom,
    ! [I5: set_int,I: int,Fa: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(Fa,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Fa,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8749_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,Fa: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(Fa,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(Fa,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8750_sum__pos2,axiom,
    ! [I5: set_Code_integer,I: code_integer,Fa: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_Code_integer_real(Fa,I))
         => ( ! [I2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(Fa,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups1270011288395367621r_real(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8751_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,Fa: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(Fa,I))
         => ( ! [I2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(Fa,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups136491112297645522BT_rat(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8752_sum__pos2,axiom,
    ! [I5: set_real,I: real,Fa: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( aa_set_real_o(member_real(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(Fa,I))
         => ( ! [I2: real] :
                ( aa_set_real_o(member_real(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(Fa,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8753_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(Fa,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Fa,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8754_sum__pos2,axiom,
    ! [I5: set_int,I: int,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(Fa,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8755_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,Fa: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(Fa,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(Fa,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(Fa,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_8756_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(Fa,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups2240296850493347238T_real(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8757_sum__pos,axiom,
    ! [I5: set_complex,Fa: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(Fa,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8758_sum__pos,axiom,
    ! [I5: set_Code_integer,Fa: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_Code_integer_real(Fa,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups1270011288395367621r_real(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8759_sum__pos,axiom,
    ! [I5: set_real,Fa: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(Fa,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8760_sum__pos,axiom,
    ! [I5: set_int,Fa: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(Fa,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8761_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,I5)
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(Fa,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups136491112297645522BT_rat(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8762_sum__pos,axiom,
    ! [I5: set_complex,Fa: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(Fa,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8763_sum__pos,axiom,
    ! [I5: set_Code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,I5)
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_Code_integer_rat(Fa,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups6602215022474089585er_rat(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8764_sum__pos,axiom,
    ! [I5: set_real,Fa: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(Fa,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8765_sum__pos,axiom,
    ! [I5: set_nat,Fa: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(Fa,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(Fa,I5)) ) ) ) ).

% sum_pos
tff(fact_8766_suminf__of__real,axiom,
    ! [X5: fun_nat_real] :
      ( summable_real(X5)
     => ( aa_real_real(real_V1803761363581548252l_real,suminf_real(X5)) = suminf_real(aTP_Lamm_agv(X5)) ) ) ).

% suminf_of_real
tff(fact_8767_suminf__of__real,axiom,
    ! [X5: fun_nat_real] :
      ( summable_real(X5)
     => ( aa_real_complex(real_V4546457046886955230omplex,suminf_real(X5)) = suminf_complex(aTP_Lamm_agg(X5)) ) ) ).

% suminf_of_real
tff(fact_8768_norm__less__p1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,aa_real_real(real_V7735802525324610683m_real,Xb))),one_one_real))) ).

% norm_less_p1
tff(fact_8769_norm__less__p1,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,Xb))),one_one_complex))) ).

% norm_less_p1
tff(fact_8770_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_VEBT_VEBT,S3: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T5)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S3),T5)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(T5,S3))
             => ( aa_VEBT_VEBT_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),S3)
               => ( aa_VEBT_VEBT_real(G,X2) = aa_VEBT_VEBT_real(H,X2) ) )
           => ( groups2240296850493347238T_real(G,T5) = groups2240296850493347238T_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8771_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S3: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T5)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T5)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T5,S3))
             => ( aa_real_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S3)
               => ( aa_real_real(G,X2) = aa_real_real(H,X2) ) )
           => ( groups8097168146408367636l_real(G,T5) = groups8097168146408367636l_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8772_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S3: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T5)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T5)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T5,S3))
             => ( aa_int_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S3)
               => ( aa_int_real(G,X2) = aa_int_real(H,X2) ) )
           => ( groups8778361861064173332t_real(G,T5) = groups8778361861064173332t_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8773_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T5)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T5,S3))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => ( aa_complex_real(G,X2) = aa_complex_real(H,X2) ) )
           => ( groups5808333547571424918x_real(G,T5) = groups5808333547571424918x_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8774_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_Code_integer,S3: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T5)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T5,S3))
             => ( aa_Code_integer_real(G,X2) = zero_zero_real ) )
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S3)
               => ( aa_Code_integer_real(G,X2) = aa_Code_integer_real(H,X2) ) )
           => ( groups1270011288395367621r_real(G,T5) = groups1270011288395367621r_real(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8775_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_VEBT_VEBT,S3: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T5)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S3),T5)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(T5,S3))
             => ( aa_VEBT_VEBT_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),S3)
               => ( aa_VEBT_VEBT_rat(G,X2) = aa_VEBT_VEBT_rat(H,X2) ) )
           => ( groups136491112297645522BT_rat(G,T5) = groups136491112297645522BT_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8776_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S3: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T5)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T5)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T5,S3))
             => ( aa_real_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S3)
               => ( aa_real_rat(G,X2) = aa_real_rat(H,X2) ) )
           => ( groups1300246762558778688al_rat(G,T5) = groups1300246762558778688al_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8777_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S3: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T5)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T5)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T5,S3))
             => ( aa_int_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S3)
               => ( aa_int_rat(G,X2) = aa_int_rat(H,X2) ) )
           => ( groups3906332499630173760nt_rat(G,T5) = groups3906332499630173760nt_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8778_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T5)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T5,S3))
             => ( aa_complex_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => ( aa_complex_rat(G,X2) = aa_complex_rat(H,X2) ) )
           => ( groups5058264527183730370ex_rat(G,T5) = groups5058264527183730370ex_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8779_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_Code_integer,S3: set_Code_integer,G: fun_Code_integer_rat,H: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T5)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T5,S3))
             => ( aa_Code_integer_rat(G,X2) = zero_zero_rat ) )
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S3)
               => ( aa_Code_integer_rat(G,X2) = aa_Code_integer_rat(H,X2) ) )
           => ( groups6602215022474089585er_rat(G,T5) = groups6602215022474089585er_rat(H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_8780_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_VEBT_VEBT,S3: set_VEBT_VEBT,H: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T5)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S3),T5)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(T5,S3))
             => ( aa_VEBT_VEBT_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),S3)
               => ( aa_VEBT_VEBT_real(G,X2) = aa_VEBT_VEBT_real(H,X2) ) )
           => ( groups2240296850493347238T_real(G,S3) = groups2240296850493347238T_real(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8781_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S3: set_real,H: fun_real_real,G: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,T5)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T5)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T5,S3))
             => ( aa_real_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S3)
               => ( aa_real_real(G,X2) = aa_real_real(H,X2) ) )
           => ( groups8097168146408367636l_real(G,S3) = groups8097168146408367636l_real(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8782_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S3: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T5)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T5)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T5,S3))
             => ( aa_int_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S3)
               => ( aa_int_real(G,X2) = aa_int_real(H,X2) ) )
           => ( groups8778361861064173332t_real(G,S3) = groups8778361861064173332t_real(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8783_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_complex,S3: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T5)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T5,S3))
             => ( aa_complex_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => ( aa_complex_real(G,X2) = aa_complex_real(H,X2) ) )
           => ( groups5808333547571424918x_real(G,S3) = groups5808333547571424918x_real(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8784_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_Code_integer,S3: set_Code_integer,H: fun_Co3960896333048504454r_real,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T5)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T5,S3))
             => ( aa_Code_integer_real(H,X2) = zero_zero_real ) )
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S3)
               => ( aa_Code_integer_real(G,X2) = aa_Code_integer_real(H,X2) ) )
           => ( groups1270011288395367621r_real(G,S3) = groups1270011288395367621r_real(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8785_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_VEBT_VEBT,S3: set_VEBT_VEBT,H: fun_VEBT_VEBT_rat,G: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,T5)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(S3),T5)
       => ( ! [X2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(T5,S3))
             => ( aa_VEBT_VEBT_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),S3)
               => ( aa_VEBT_VEBT_rat(G,X2) = aa_VEBT_VEBT_rat(H,X2) ) )
           => ( groups136491112297645522BT_rat(G,S3) = groups136491112297645522BT_rat(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8786_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S3: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,T5)
     => ( aa_set_real_o(ord_less_eq_set_real(S3),T5)
       => ( ! [X2: real] :
              ( aa_set_real_o(member_real(X2),minus_minus_set_real(T5,S3))
             => ( aa_real_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),S3)
               => ( aa_real_rat(G,X2) = aa_real_rat(H,X2) ) )
           => ( groups1300246762558778688al_rat(G,S3) = groups1300246762558778688al_rat(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8787_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S3: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T5)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T5)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T5,S3))
             => ( aa_int_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: int] :
                ( aa_set_int_o(member_int(X2),S3)
               => ( aa_int_rat(G,X2) = aa_int_rat(H,X2) ) )
           => ( groups3906332499630173760nt_rat(G,S3) = groups3906332499630173760nt_rat(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8788_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_complex,S3: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T5)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T5,S3))
             => ( aa_complex_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: complex] :
                ( aa_set_complex_o(member_complex(X2),S3)
               => ( aa_complex_rat(G,X2) = aa_complex_rat(H,X2) ) )
           => ( groups5058264527183730370ex_rat(G,S3) = groups5058264527183730370ex_rat(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8789_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_Code_integer,S3: set_Code_integer,H: fun_Code_integer_rat,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T5)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T5,S3))
             => ( aa_Code_integer_rat(H,X2) = zero_zero_rat ) )
         => ( ! [X2: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X2),S3)
               => ( aa_Code_integer_rat(G,X2) = aa_Code_integer_rat(H,X2) ) )
           => ( groups6602215022474089585er_rat(G,S3) = groups6602215022474089585er_rat(H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_8790_sum_Omono__neutral__right,axiom,
    ! [T5: set_int,S3: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T5)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T5)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T5,S3))
             => ( aa_int_real(G,X2) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,T5) = groups8778361861064173332t_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8791_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T5)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T5,S3))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T5) = groups5808333547571424918x_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8792_sum_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S3: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T5)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T5,S3))
             => ( aa_Code_integer_real(G,X2) = zero_zero_real ) )
         => ( groups1270011288395367621r_real(G,T5) = groups1270011288395367621r_real(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8793_sum_Omono__neutral__right,axiom,
    ! [T5: set_int,S3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T5)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T5)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T5,S3))
             => ( aa_int_rat(G,X2) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,T5) = groups3906332499630173760nt_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8794_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T5)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T5,S3))
             => ( aa_complex_rat(G,X2) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T5) = groups5058264527183730370ex_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8795_sum_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T5)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T5,S3))
             => ( aa_Code_integer_rat(G,X2) = zero_zero_rat ) )
         => ( groups6602215022474089585er_rat(G,T5) = groups6602215022474089585er_rat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8796_sum_Omono__neutral__right,axiom,
    ! [T5: set_int,S3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T5)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T5)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T5,S3))
             => ( aa_int_nat(G,X2) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,T5) = groups4541462559716669496nt_nat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8797_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T5)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T5,S3))
             => ( aa_complex_nat(G,X2) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T5) = groups5693394587270226106ex_nat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8798_sum_Omono__neutral__right,axiom,
    ! [T5: set_Code_integer,S3: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T5)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T5,S3))
             => ( aa_Code_integer_nat(G,X2) = zero_zero_nat ) )
         => ( groups7237345082560585321er_nat(G,T5) = groups7237345082560585321er_nat(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8799_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S3: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T5)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T5,S3))
             => ( aa_complex_int(G,X2) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T5) = groups5690904116761175830ex_int(G,S3) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_8800_sum_Omono__neutral__left,axiom,
    ! [T5: set_int,S3: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T5)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T5)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T5,S3))
             => ( aa_int_real(G,X2) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,S3) = groups8778361861064173332t_real(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8801_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T5)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T5,S3))
             => ( aa_complex_real(G,X2) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S3) = groups5808333547571424918x_real(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8802_sum_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S3: set_Code_integer,G: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T5)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T5,S3))
             => ( aa_Code_integer_real(G,X2) = zero_zero_real ) )
         => ( groups1270011288395367621r_real(G,S3) = groups1270011288395367621r_real(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8803_sum_Omono__neutral__left,axiom,
    ! [T5: set_int,S3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T5)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T5)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T5,S3))
             => ( aa_int_rat(G,X2) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,S3) = groups3906332499630173760nt_rat(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8804_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S3: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T5)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T5,S3))
             => ( aa_complex_rat(G,X2) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S3) = groups5058264527183730370ex_rat(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8805_sum_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S3: set_Code_integer,G: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T5)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T5,S3))
             => ( aa_Code_integer_rat(G,X2) = zero_zero_rat ) )
         => ( groups6602215022474089585er_rat(G,S3) = groups6602215022474089585er_rat(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8806_sum_Omono__neutral__left,axiom,
    ! [T5: set_int,S3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T5)
     => ( aa_set_int_o(ord_less_eq_set_int(S3),T5)
       => ( ! [X2: int] :
              ( aa_set_int_o(member_int(X2),minus_minus_set_int(T5,S3))
             => ( aa_int_nat(G,X2) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,S3) = groups4541462559716669496nt_nat(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8807_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S3: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T5)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T5,S3))
             => ( aa_complex_nat(G,X2) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S3) = groups5693394587270226106ex_nat(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8808_sum_Omono__neutral__left,axiom,
    ! [T5: set_Code_integer,S3: set_Code_integer,G: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S3),T5)
       => ( ! [X2: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(T5,S3))
             => ( aa_Code_integer_nat(G,X2) = zero_zero_nat ) )
         => ( groups7237345082560585321er_nat(G,S3) = groups7237345082560585321er_nat(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8809_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S3: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S3),T5)
       => ( ! [X2: complex] :
              ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(T5,S3))
             => ( aa_complex_int(G,X2) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S3) = groups5690904116761175830ex_int(G,T5) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_8810_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A3: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A3),C4)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B5),C4)
         => ( ! [A: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(C4,A3))
               => ( aa_VEBT_VEBT_real(G,A) = zero_zero_real ) )
           => ( ! [B: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(C4,B5))
                 => ( aa_VEBT_VEBT_real(H,B) = zero_zero_real ) )
             => ( ( groups2240296850493347238T_real(G,C4) = groups2240296850493347238T_real(H,C4) )
               => ( groups2240296850493347238T_real(G,A3) = groups2240296850493347238T_real(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8811_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A3: set_real,B5: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A3),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B5),C4)
         => ( ! [A: real] :
                ( aa_set_real_o(member_real(A),minus_minus_set_real(C4,A3))
               => ( aa_real_real(G,A) = zero_zero_real ) )
           => ( ! [B: real] :
                  ( aa_set_real_o(member_real(B),minus_minus_set_real(C4,B5))
                 => ( aa_real_real(H,B) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) )
               => ( groups8097168146408367636l_real(G,A3) = groups8097168146408367636l_real(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8812_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A3: set_int,B5: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,C4)
     => ( aa_set_int_o(ord_less_eq_set_int(A3),C4)
       => ( aa_set_int_o(ord_less_eq_set_int(B5),C4)
         => ( ! [A: int] :
                ( aa_set_int_o(member_int(A),minus_minus_set_int(C4,A3))
               => ( aa_int_real(G,A) = zero_zero_real ) )
           => ( ! [B: int] :
                  ( aa_set_int_o(member_int(B),minus_minus_set_int(C4,B5))
                 => ( aa_int_real(H,B) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,C4) = groups8778361861064173332t_real(H,C4) )
               => ( groups8778361861064173332t_real(G,A3) = groups8778361861064173332t_real(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8813_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A3: set_complex,B5: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B5),C4)
         => ( ! [A: complex] :
                ( aa_set_complex_o(member_complex(A),minus_811609699411566653omplex(C4,A3))
               => ( aa_complex_real(G,A) = zero_zero_real ) )
           => ( ! [B: complex] :
                  ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(C4,B5))
                 => ( aa_complex_real(H,B) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) )
               => ( groups5808333547571424918x_real(G,A3) = groups5808333547571424918x_real(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8814_sum_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A3: set_Code_integer,B5: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C4)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B5),C4)
         => ( ! [A: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A),minus_2355218937544613996nteger(C4,A3))
               => ( aa_Code_integer_real(G,A) = zero_zero_real ) )
           => ( ! [B: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(C4,B5))
                 => ( aa_Code_integer_real(H,B) = zero_zero_real ) )
             => ( ( groups1270011288395367621r_real(G,C4) = groups1270011288395367621r_real(H,C4) )
               => ( groups1270011288395367621r_real(G,A3) = groups1270011288395367621r_real(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8815_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A3: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A3),C4)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B5),C4)
         => ( ! [A: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(C4,A3))
               => ( aa_VEBT_VEBT_rat(G,A) = zero_zero_rat ) )
           => ( ! [B: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(C4,B5))
                 => ( aa_VEBT_VEBT_rat(H,B) = zero_zero_rat ) )
             => ( ( groups136491112297645522BT_rat(G,C4) = groups136491112297645522BT_rat(H,C4) )
               => ( groups136491112297645522BT_rat(G,A3) = groups136491112297645522BT_rat(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8816_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A3: set_real,B5: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A3),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B5),C4)
         => ( ! [A: real] :
                ( aa_set_real_o(member_real(A),minus_minus_set_real(C4,A3))
               => ( aa_real_rat(G,A) = zero_zero_rat ) )
           => ( ! [B: real] :
                  ( aa_set_real_o(member_real(B),minus_minus_set_real(C4,B5))
                 => ( aa_real_rat(H,B) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) )
               => ( groups1300246762558778688al_rat(G,A3) = groups1300246762558778688al_rat(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8817_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A3: set_int,B5: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,C4)
     => ( aa_set_int_o(ord_less_eq_set_int(A3),C4)
       => ( aa_set_int_o(ord_less_eq_set_int(B5),C4)
         => ( ! [A: int] :
                ( aa_set_int_o(member_int(A),minus_minus_set_int(C4,A3))
               => ( aa_int_rat(G,A) = zero_zero_rat ) )
           => ( ! [B: int] :
                  ( aa_set_int_o(member_int(B),minus_minus_set_int(C4,B5))
                 => ( aa_int_rat(H,B) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,C4) = groups3906332499630173760nt_rat(H,C4) )
               => ( groups3906332499630173760nt_rat(G,A3) = groups3906332499630173760nt_rat(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8818_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A3: set_complex,B5: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B5),C4)
         => ( ! [A: complex] :
                ( aa_set_complex_o(member_complex(A),minus_811609699411566653omplex(C4,A3))
               => ( aa_complex_rat(G,A) = zero_zero_rat ) )
           => ( ! [B: complex] :
                  ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(C4,B5))
                 => ( aa_complex_rat(H,B) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) )
               => ( groups5058264527183730370ex_rat(G,A3) = groups5058264527183730370ex_rat(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8819_sum_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A3: set_Code_integer,B5: set_Code_integer,G: fun_Code_integer_rat,H: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C4)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B5),C4)
         => ( ! [A: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A),minus_2355218937544613996nteger(C4,A3))
               => ( aa_Code_integer_rat(G,A) = zero_zero_rat ) )
           => ( ! [B: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(C4,B5))
                 => ( aa_Code_integer_rat(H,B) = zero_zero_rat ) )
             => ( ( groups6602215022474089585er_rat(G,C4) = groups6602215022474089585er_rat(H,C4) )
               => ( groups6602215022474089585er_rat(G,A3) = groups6602215022474089585er_rat(H,B5) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_8820_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A3: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: fun_VEBT_VEBT_real,H: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A3),C4)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B5),C4)
         => ( ! [A: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(C4,A3))
               => ( aa_VEBT_VEBT_real(G,A) = zero_zero_real ) )
           => ( ! [B: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(C4,B5))
                 => ( aa_VEBT_VEBT_real(H,B) = zero_zero_real ) )
             => ( ( groups2240296850493347238T_real(G,A3) = groups2240296850493347238T_real(H,B5) )
              <=> ( groups2240296850493347238T_real(G,C4) = groups2240296850493347238T_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8821_sum_Osame__carrier,axiom,
    ! [C4: set_real,A3: set_real,B5: set_real,G: fun_real_real,H: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A3),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B5),C4)
         => ( ! [A: real] :
                ( aa_set_real_o(member_real(A),minus_minus_set_real(C4,A3))
               => ( aa_real_real(G,A) = zero_zero_real ) )
           => ( ! [B: real] :
                  ( aa_set_real_o(member_real(B),minus_minus_set_real(C4,B5))
                 => ( aa_real_real(H,B) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A3) = groups8097168146408367636l_real(H,B5) )
              <=> ( groups8097168146408367636l_real(G,C4) = groups8097168146408367636l_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8822_sum_Osame__carrier,axiom,
    ! [C4: set_int,A3: set_int,B5: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,C4)
     => ( aa_set_int_o(ord_less_eq_set_int(A3),C4)
       => ( aa_set_int_o(ord_less_eq_set_int(B5),C4)
         => ( ! [A: int] :
                ( aa_set_int_o(member_int(A),minus_minus_set_int(C4,A3))
               => ( aa_int_real(G,A) = zero_zero_real ) )
           => ( ! [B: int] :
                  ( aa_set_int_o(member_int(B),minus_minus_set_int(C4,B5))
                 => ( aa_int_real(H,B) = zero_zero_real ) )
             => ( ( groups8778361861064173332t_real(G,A3) = groups8778361861064173332t_real(H,B5) )
              <=> ( groups8778361861064173332t_real(G,C4) = groups8778361861064173332t_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8823_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A3: set_complex,B5: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B5),C4)
         => ( ! [A: complex] :
                ( aa_set_complex_o(member_complex(A),minus_811609699411566653omplex(C4,A3))
               => ( aa_complex_real(G,A) = zero_zero_real ) )
           => ( ! [B: complex] :
                  ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(C4,B5))
                 => ( aa_complex_real(H,B) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A3) = groups5808333547571424918x_real(H,B5) )
              <=> ( groups5808333547571424918x_real(G,C4) = groups5808333547571424918x_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8824_sum_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A3: set_Code_integer,B5: set_Code_integer,G: fun_Co3960896333048504454r_real,H: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C4)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B5),C4)
         => ( ! [A: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A),minus_2355218937544613996nteger(C4,A3))
               => ( aa_Code_integer_real(G,A) = zero_zero_real ) )
           => ( ! [B: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(C4,B5))
                 => ( aa_Code_integer_real(H,B) = zero_zero_real ) )
             => ( ( groups1270011288395367621r_real(G,A3) = groups1270011288395367621r_real(H,B5) )
              <=> ( groups1270011288395367621r_real(G,C4) = groups1270011288395367621r_real(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8825_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A3: set_VEBT_VEBT,B5: set_VEBT_VEBT,G: fun_VEBT_VEBT_rat,H: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,C4)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A3),C4)
       => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B5),C4)
         => ( ! [A: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A),minus_5127226145743854075T_VEBT(C4,A3))
               => ( aa_VEBT_VEBT_rat(G,A) = zero_zero_rat ) )
           => ( ! [B: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(C4,B5))
                 => ( aa_VEBT_VEBT_rat(H,B) = zero_zero_rat ) )
             => ( ( groups136491112297645522BT_rat(G,A3) = groups136491112297645522BT_rat(H,B5) )
              <=> ( groups136491112297645522BT_rat(G,C4) = groups136491112297645522BT_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8826_sum_Osame__carrier,axiom,
    ! [C4: set_real,A3: set_real,B5: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,C4)
     => ( aa_set_real_o(ord_less_eq_set_real(A3),C4)
       => ( aa_set_real_o(ord_less_eq_set_real(B5),C4)
         => ( ! [A: real] :
                ( aa_set_real_o(member_real(A),minus_minus_set_real(C4,A3))
               => ( aa_real_rat(G,A) = zero_zero_rat ) )
           => ( ! [B: real] :
                  ( aa_set_real_o(member_real(B),minus_minus_set_real(C4,B5))
                 => ( aa_real_rat(H,B) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A3) = groups1300246762558778688al_rat(H,B5) )
              <=> ( groups1300246762558778688al_rat(G,C4) = groups1300246762558778688al_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8827_sum_Osame__carrier,axiom,
    ! [C4: set_int,A3: set_int,B5: set_int,G: fun_int_rat,H: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,C4)
     => ( aa_set_int_o(ord_less_eq_set_int(A3),C4)
       => ( aa_set_int_o(ord_less_eq_set_int(B5),C4)
         => ( ! [A: int] :
                ( aa_set_int_o(member_int(A),minus_minus_set_int(C4,A3))
               => ( aa_int_rat(G,A) = zero_zero_rat ) )
           => ( ! [B: int] :
                  ( aa_set_int_o(member_int(B),minus_minus_set_int(C4,B5))
                 => ( aa_int_rat(H,B) = zero_zero_rat ) )
             => ( ( groups3906332499630173760nt_rat(G,A3) = groups3906332499630173760nt_rat(H,B5) )
              <=> ( groups3906332499630173760nt_rat(G,C4) = groups3906332499630173760nt_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8828_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A3: set_complex,B5: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C4)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B5),C4)
         => ( ! [A: complex] :
                ( aa_set_complex_o(member_complex(A),minus_811609699411566653omplex(C4,A3))
               => ( aa_complex_rat(G,A) = zero_zero_rat ) )
           => ( ! [B: complex] :
                  ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(C4,B5))
                 => ( aa_complex_rat(H,B) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,A3) = groups5058264527183730370ex_rat(H,B5) )
              <=> ( groups5058264527183730370ex_rat(G,C4) = groups5058264527183730370ex_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8829_sum_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A3: set_Code_integer,B5: set_Code_integer,G: fun_Code_integer_rat,H: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,C4)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),C4)
       => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(B5),C4)
         => ( ! [A: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(A),minus_2355218937544613996nteger(C4,A3))
               => ( aa_Code_integer_rat(G,A) = zero_zero_rat ) )
           => ( ! [B: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(C4,B5))
                 => ( aa_Code_integer_rat(H,B) = zero_zero_rat ) )
             => ( ( groups6602215022474089585er_rat(G,A3) = groups6602215022474089585er_rat(H,B5) )
              <=> ( groups6602215022474089585er_rat(G,C4) = groups6602215022474089585er_rat(H,C4) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_8830_sum__shift__lb__Suc0__0,axiom,
    ! [Fa: fun_na2984781812408219459l_num1,K: nat] :
      ( ( aa_nat5599461682457136004l_num1(Fa,zero_zero_nat) = zero_z3563351764282998399l_num1 )
     => ( groups2996710295995929986l_num1(Fa,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),K)) = groups2996710295995929986l_num1(Fa,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8831_sum__shift__lb__Suc0__0,axiom,
    ! [Fa: fun_nat_rat,K: nat] :
      ( ( aa_nat_rat(Fa,zero_zero_nat) = zero_zero_rat )
     => ( groups2906978787729119204at_rat(Fa,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),K)) = groups2906978787729119204at_rat(Fa,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8832_sum__shift__lb__Suc0__0,axiom,
    ! [Fa: fun_nat_int,K: nat] :
      ( ( aa_nat_int(Fa,zero_zero_nat) = zero_zero_int )
     => ( groups3539618377306564664at_int(Fa,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),K)) = groups3539618377306564664at_int(Fa,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8833_sum__shift__lb__Suc0__0,axiom,
    ! [Fa: fun_nat_nat,K: nat] :
      ( ( aa_nat_nat(Fa,zero_zero_nat) = zero_zero_nat )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(Fa),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(Fa),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8834_sum__shift__lb__Suc0__0,axiom,
    ! [Fa: fun_nat_real,K: nat] :
      ( ( aa_nat_real(Fa,zero_zero_nat) = zero_zero_real )
     => ( groups6591440286371151544t_real(Fa,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),K)) = groups6591440286371151544t_real(Fa,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_8835_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,N: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)),aa_nat_rat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8836_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,N: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))),aa_nat_int(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8837_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8838_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,N: nat] : groups6591440286371151544t_real(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))),aa_nat_real(G,aa_nat_nat(suc,N))) ).

% sum.atLeast0_atMost_Suc
tff(fact_8839_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) = plus_plus_rat(aa_nat_rat(G,M),groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8840_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8841_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8842_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),groups6591440286371151544t_real(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,M)),N))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_8843_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N))) = plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,N)),groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8844_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,N))),groups3539618377306564664at_int(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8845_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8846_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,aa_nat_nat(suc,N))),groups6591440286371151544t_real(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_8847_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)),aa_nat_rat(G,aa_nat_nat(suc,N))) = plus_plus_rat(aa_nat_rat(G,M),groups2906978787729119204at_rat(aTP_Lamm_anj(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8848_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))),aa_nat_int(G,aa_nat_nat(suc,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(aTP_Lamm_ank(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8849_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))),aa_nat_nat(G,aa_nat_nat(suc,N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_amg(G)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8850_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))),aa_nat_real(G,aa_nat_nat(suc,N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),groups6591440286371151544t_real(aTP_Lamm_vl(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_8851_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,Fa: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( groups2906978787729119204at_rat(aTP_Lamm_anl(Fa),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) = minus_minus_rat(aa_nat_rat(Fa,aa_nat_nat(suc,N)),aa_nat_rat(Fa,M)) ) ) ).

% sum_Suc_diff
tff(fact_8852_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,Fa: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( groups3539618377306564664at_int(aTP_Lamm_anm(Fa),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Fa,aa_nat_nat(suc,N))),aa_nat_int(Fa,M)) ) ) ).

% sum_Suc_diff
tff(fact_8853_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,Fa: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,N))
     => ( groups6591440286371151544t_real(aTP_Lamm_ann(Fa),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) = aa_real_real(minus_minus_real(aa_nat_real(Fa,aa_nat_nat(suc,N))),aa_nat_real(Fa,M)) ) ) ).

% sum_Suc_diff
tff(fact_8854_sum__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A3: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B5)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A3),B5)
       => ( ! [B: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(B5,A3))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(Fa,B)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2240296850493347238T_real(Fa,A3)),groups2240296850493347238T_real(Fa,B5)) ) ) ) ).

% sum_mono2
tff(fact_8855_sum__mono2,axiom,
    ! [B5: set_real,A3: set_real,Fa: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,B5)
     => ( aa_set_real_o(ord_less_eq_set_real(A3),B5)
       => ( ! [B: real] :
              ( aa_set_real_o(member_real(B),minus_minus_set_real(B5,A3))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Fa,B)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(Fa,A3)),groups8097168146408367636l_real(Fa,B5)) ) ) ) ).

% sum_mono2
tff(fact_8856_sum__mono2,axiom,
    ! [B5: set_int,A3: set_int,Fa: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,B5)
     => ( aa_set_int_o(ord_less_eq_set_int(A3),B5)
       => ( ! [B: int] :
              ( aa_set_int_o(member_int(B),minus_minus_set_int(B5,A3))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Fa,B)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(Fa,A3)),groups8778361861064173332t_real(Fa,B5)) ) ) ) ).

% sum_mono2
tff(fact_8857_sum__mono2,axiom,
    ! [B5: set_complex,A3: set_complex,Fa: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B5)
       => ( ! [B: complex] :
              ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B5,A3))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(Fa,B)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(Fa,A3)),groups5808333547571424918x_real(Fa,B5)) ) ) ) ).

% sum_mono2
tff(fact_8858_sum__mono2,axiom,
    ! [B5: set_Code_integer,A3: set_Code_integer,Fa: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B5)
       => ( ! [B: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(B5,A3))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(Fa,B)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1270011288395367621r_real(Fa,A3)),groups1270011288395367621r_real(Fa,B5)) ) ) ) ).

% sum_mono2
tff(fact_8859_sum__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A3: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B5)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A3),B5)
       => ( ! [B: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B),minus_5127226145743854075T_VEBT(B5,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(Fa,B)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups136491112297645522BT_rat(Fa,A3)),groups136491112297645522BT_rat(Fa,B5)) ) ) ) ).

% sum_mono2
tff(fact_8860_sum__mono2,axiom,
    ! [B5: set_real,A3: set_real,Fa: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,B5)
     => ( aa_set_real_o(ord_less_eq_set_real(A3),B5)
       => ( ! [B: real] :
              ( aa_set_real_o(member_real(B),minus_minus_set_real(B5,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(Fa,B)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(Fa,A3)),groups1300246762558778688al_rat(Fa,B5)) ) ) ) ).

% sum_mono2
tff(fact_8861_sum__mono2,axiom,
    ! [B5: set_int,A3: set_int,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,B5)
     => ( aa_set_int_o(ord_less_eq_set_int(A3),B5)
       => ( ! [B: int] :
              ( aa_set_int_o(member_int(B),minus_minus_set_int(B5,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,B)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(Fa,A3)),groups3906332499630173760nt_rat(Fa,B5)) ) ) ) ).

% sum_mono2
tff(fact_8862_sum__mono2,axiom,
    ! [B5: set_complex,A3: set_complex,Fa: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B5)
       => ( ! [B: complex] :
              ( aa_set_complex_o(member_complex(B),minus_811609699411566653omplex(B5,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(Fa,B)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(Fa,A3)),groups5058264527183730370ex_rat(Fa,B5)) ) ) ) ).

% sum_mono2
tff(fact_8863_sum__mono2,axiom,
    ! [B5: set_Code_integer,A3: set_Code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B5)
       => ( ! [B: code_integer] :
              ( aa_set5643387332196575111eger_o(member_Code_integer(B),minus_2355218937544613996nteger(B5,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(Fa,B)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups6602215022474089585er_rat(Fa,A3)),groups6602215022474089585er_rat(Fa,B5)) ) ) ) ).

% sum_mono2
tff(fact_8864_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)) ) ) ).

% arccos_lbound
tff(fact_8865_arccos__less__arccos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_less_arccos
tff(fact_8866_arccos__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Xb)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ).

% arccos_less_mono
tff(fact_8867_arccos__cos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xb)) = Xb ) ) ) ).

% arccos_cos
tff(fact_8868_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_rat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)),groups2906978787729119204at_rat(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_8869_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_int,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))),groups3539618377306564664at_int(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_8870_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_nat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_8871_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: fun_nat_real,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))
     => ( groups6591440286371151544t_real(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N))),groups6591440286371151544t_real(G,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_8872_sum__le__suminf,axiom,
    ! [Fa: fun_nat_int,I5: set_nat] :
      ( summable_int(Fa)
     => ( aa_set_nat_o(finite_finite_nat,I5)
       => ( ! [N3: nat] :
              ( aa_set_nat_o(member_nat(N3),uminus5710092332889474511et_nat(I5))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(Fa,N3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(Fa,I5)),suminf_int(Fa)) ) ) ) ).

% sum_le_suminf
tff(fact_8873_sum__le__suminf,axiom,
    ! [Fa: fun_nat_nat,I5: set_nat] :
      ( summable_nat(Fa)
     => ( aa_set_nat_o(finite_finite_nat,I5)
       => ( ! [N3: nat] :
              ( aa_set_nat_o(member_nat(N3),uminus5710092332889474511et_nat(I5))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(Fa,N3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Fa),I5)),suminf_nat(Fa)) ) ) ) ).

% sum_le_suminf
tff(fact_8874_sum__le__suminf,axiom,
    ! [Fa: fun_nat_real,I5: set_nat] :
      ( summable_real(Fa)
     => ( aa_set_nat_o(finite_finite_nat,I5)
       => ( ! [N3: nat] :
              ( aa_set_nat_o(member_nat(N3),uminus5710092332889474511et_nat(I5))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Fa,N3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(Fa,I5)),suminf_real(Fa)) ) ) ) ).

% sum_le_suminf
tff(fact_8875_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_real,C: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups2240296850493347238T_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_ano(A2),B2),C),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),S3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(B2,A2)),groups2240296850493347238T_real(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)))),groups2240296850493347238T_real(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.delta_remove
tff(fact_8876_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A2: complex,B2: fun_complex_real,C: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_anp(A2),B2),C),S3) = $ite(aa_set_complex_o(member_complex(A2),S3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(B2,A2)),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S3,insert_complex(A2,bot_bot_set_complex)))),groups5808333547571424918x_real(C,minus_811609699411566653omplex(S3,insert_complex(A2,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_8877_sum_Odelta__remove,axiom,
    ! [S3: set_Code_integer,A2: code_integer,B2: fun_Co3960896333048504454r_real,C: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( groups1270011288395367621r_real(aa_fun6341089502340155082r_real(aa_fun7644355603919855821r_real(aTP_Lamm_anq(A2),B2),C),S3) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A2),S3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(B2,A2)),groups1270011288395367621r_real(C,minus_2355218937544613996nteger(S3,insert_Code_integer(A2,bot_bo3990330152332043303nteger)))),groups1270011288395367621r_real(C,minus_2355218937544613996nteger(S3,insert_Code_integer(A2,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.delta_remove
tff(fact_8878_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_rat,C: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups136491112297645522BT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_anr(A2),B2),C),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),S3),plus_plus_rat(aa_VEBT_VEBT_rat(B2,A2),groups136491112297645522BT_rat(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)))),groups136491112297645522BT_rat(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.delta_remove
tff(fact_8879_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A2: complex,B2: fun_complex_rat,C: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_ans(A2),B2),C),S3) = $ite(aa_set_complex_o(member_complex(A2),S3),plus_plus_rat(aa_complex_rat(B2,A2),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S3,insert_complex(A2,bot_bot_set_complex)))),groups5058264527183730370ex_rat(C,minus_811609699411566653omplex(S3,insert_complex(A2,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_8880_sum_Odelta__remove,axiom,
    ! [S3: set_Code_integer,A2: code_integer,B2: fun_Code_integer_rat,C: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( groups6602215022474089585er_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_ant(A2),B2),C),S3) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A2),S3),plus_plus_rat(aa_Code_integer_rat(B2,A2),groups6602215022474089585er_rat(C,minus_2355218937544613996nteger(S3,insert_Code_integer(A2,bot_bo3990330152332043303nteger)))),groups6602215022474089585er_rat(C,minus_2355218937544613996nteger(S3,insert_Code_integer(A2,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.delta_remove
tff(fact_8881_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_nat,C: fun_VEBT_VEBT_nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups771621172384141258BT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_anu(A2),B2),C),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),S3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(B2,A2)),groups771621172384141258BT_nat(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)))),groups771621172384141258BT_nat(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.delta_remove
tff(fact_8882_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A2: complex,B2: fun_complex_nat,C: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_anv(A2),B2),C),S3) = $ite(aa_set_complex_o(member_complex(A2),S3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_complex_nat(B2,A2)),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S3,insert_complex(A2,bot_bot_set_complex)))),groups5693394587270226106ex_nat(C,minus_811609699411566653omplex(S3,insert_complex(A2,bot_bot_set_complex)))) ) ) ).

% sum.delta_remove
tff(fact_8883_sum_Odelta__remove,axiom,
    ! [S3: set_Code_integer,A2: code_integer,B2: fun_Code_integer_nat,C: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S3)
     => ( groups7237345082560585321er_nat(aa_fun3076867170244285586er_nat(aa_fun7635586485349294009er_nat(aTP_Lamm_anw(A2),B2),C),S3) = $ite(aa_set5643387332196575111eger_o(member_Code_integer(A2),S3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Code_integer_nat(B2,A2)),groups7237345082560585321er_nat(C,minus_2355218937544613996nteger(S3,insert_Code_integer(A2,bot_bo3990330152332043303nteger)))),groups7237345082560585321er_nat(C,minus_2355218937544613996nteger(S3,insert_Code_integer(A2,bot_bo3990330152332043303nteger)))) ) ) ).

% sum.delta_remove
tff(fact_8884_sum_Odelta__remove,axiom,
    ! [S3: set_VEBT_VEBT,A2: vEBT_VEBT,B2: fun_VEBT_VEBT_int,C: fun_VEBT_VEBT_int] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S3)
     => ( groups769130701875090982BT_int(aa_fun7144222995531728252BT_int(aa_fun7485955581636121530BT_int(aTP_Lamm_anx(A2),B2),C),S3) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),S3),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_VEBT_VEBT_int(B2,A2)),groups769130701875090982BT_int(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)))),groups769130701875090982BT_int(C,minus_5127226145743854075T_VEBT(S3,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)))) ) ) ).

% sum.delta_remove
tff(fact_8885_set__encode__def,axiom,
    nat_set_encode = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% set_encode_def
tff(fact_8886_sum__strict__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A3: set_VEBT_VEBT,B2: vEBT_VEBT,Fa: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B5)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A3),B5)
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(B5,A3))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_VEBT_VEBT_real(Fa,B2))
           => ( ! [X2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),B5)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(Fa,X2)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups2240296850493347238T_real(Fa,A3)),groups2240296850493347238T_real(Fa,B5)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8887_sum__strict__mono2,axiom,
    ! [B5: set_real,A3: set_real,B2: real,Fa: fun_real_real] :
      ( aa_set_real_o(finite_finite_real,B5)
     => ( aa_set_real_o(ord_less_eq_set_real(A3),B5)
       => ( aa_set_real_o(member_real(B2),minus_minus_set_real(B5,A3))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(Fa,B2))
           => ( ! [X2: real] :
                  ( aa_set_real_o(member_real(X2),B5)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Fa,X2)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(Fa,A3)),groups8097168146408367636l_real(Fa,B5)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8888_sum__strict__mono2,axiom,
    ! [B5: set_int,A3: set_int,B2: int,Fa: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,B5)
     => ( aa_set_int_o(ord_less_eq_set_int(A3),B5)
       => ( aa_set_int_o(member_int(B2),minus_minus_set_int(B5,A3))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(Fa,B2))
           => ( ! [X2: int] :
                  ( aa_set_int_o(member_int(X2),B5)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Fa,X2)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(Fa,A3)),groups8778361861064173332t_real(Fa,B5)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8889_sum__strict__mono2,axiom,
    ! [B5: set_complex,A3: set_complex,B2: complex,Fa: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B5)
       => ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(B5,A3))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(Fa,B2))
           => ( ! [X2: complex] :
                  ( aa_set_complex_o(member_complex(X2),B5)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(Fa,X2)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(Fa,A3)),groups5808333547571424918x_real(Fa,B5)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8890_sum__strict__mono2,axiom,
    ! [B5: set_Code_integer,A3: set_Code_integer,B2: code_integer,Fa: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B5)
       => ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(B5,A3))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_Code_integer_real(Fa,B2))
           => ( ! [X2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X2),B5)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(Fa,X2)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups1270011288395367621r_real(Fa,A3)),groups1270011288395367621r_real(Fa,B5)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8891_sum__strict__mono2,axiom,
    ! [B5: set_VEBT_VEBT,A3: set_VEBT_VEBT,B2: vEBT_VEBT,Fa: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B5)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A3),B5)
       => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),minus_5127226145743854075T_VEBT(B5,A3))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_VEBT_VEBT_rat(Fa,B2))
           => ( ! [X2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),B5)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(Fa,X2)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups136491112297645522BT_rat(Fa,A3)),groups136491112297645522BT_rat(Fa,B5)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8892_sum__strict__mono2,axiom,
    ! [B5: set_real,A3: set_real,B2: real,Fa: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,B5)
     => ( aa_set_real_o(ord_less_eq_set_real(A3),B5)
       => ( aa_set_real_o(member_real(B2),minus_minus_set_real(B5,A3))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(Fa,B2))
           => ( ! [X2: real] :
                  ( aa_set_real_o(member_real(X2),B5)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(Fa,X2)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(Fa,A3)),groups1300246762558778688al_rat(Fa,B5)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8893_sum__strict__mono2,axiom,
    ! [B5: set_int,A3: set_int,B2: int,Fa: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,B5)
     => ( aa_set_int_o(ord_less_eq_set_int(A3),B5)
       => ( aa_set_int_o(member_int(B2),minus_minus_set_int(B5,A3))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(Fa,B2))
           => ( ! [X2: int] :
                  ( aa_set_int_o(member_int(X2),B5)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,X2)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(Fa,A3)),groups3906332499630173760nt_rat(Fa,B5)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8894_sum__strict__mono2,axiom,
    ! [B5: set_complex,A3: set_complex,B2: complex,Fa: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B5)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B5)
       => ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(B5,A3))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(Fa,B2))
           => ( ! [X2: complex] :
                  ( aa_set_complex_o(member_complex(X2),B5)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(Fa,X2)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(Fa,A3)),groups5058264527183730370ex_rat(Fa,B5)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8895_sum__strict__mono2,axiom,
    ! [B5: set_Code_integer,A3: set_Code_integer,B2: code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B5)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A3),B5)
       => ( aa_set5643387332196575111eger_o(member_Code_integer(B2),minus_2355218937544613996nteger(B5,A3))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_Code_integer_rat(Fa,B2))
           => ( ! [X2: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(X2),B5)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(Fa,X2)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups6602215022474089585er_rat(Fa,A3)),groups6602215022474089585er_rat(Fa,B5)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_8896_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A3: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_real] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),A3)
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(Fa,X2)) )
       => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(Fa,I)),groups2240296850493347238T_real(Fa,A3)) ) ) ) ).

% member_le_sum
tff(fact_8897_member__le__sum,axiom,
    ! [I: complex,A3: set_complex,Fa: fun_complex_real] :
      ( aa_set_complex_o(member_complex(I),A3)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(A3,insert_complex(I,bot_bot_set_complex)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(Fa,X2)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(Fa,I)),groups5808333547571424918x_real(Fa,A3)) ) ) ) ).

% member_le_sum
tff(fact_8898_member__le__sum,axiom,
    ! [I: code_integer,A3: set_Code_integer,Fa: fun_Co3960896333048504454r_real] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I),A3)
     => ( ! [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(A3,insert_Code_integer(I,bot_bo3990330152332043303nteger)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Code_integer_real(Fa,X2)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Code_integer_real(Fa,I)),groups1270011288395367621r_real(Fa,A3)) ) ) ) ).

% member_le_sum
tff(fact_8899_member__le__sum,axiom,
    ! [I: real,A3: set_real,Fa: fun_real_real] :
      ( aa_set_real_o(member_real(I),A3)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),minus_minus_set_real(A3,insert_real(I,bot_bot_set_real)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Fa,X2)) )
       => ( aa_set_real_o(finite_finite_real,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(Fa,I)),groups8097168146408367636l_real(Fa,A3)) ) ) ) ).

% member_le_sum
tff(fact_8900_member__le__sum,axiom,
    ! [I: int,A3: set_int,Fa: fun_int_real] :
      ( aa_set_int_o(member_int(I),A3)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),minus_minus_set_int(A3,insert_int(I,bot_bot_set_int)))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Fa,X2)) )
       => ( aa_set_int_o(finite_finite_int,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(Fa,I)),groups8778361861064173332t_real(Fa,A3)) ) ) ) ).

% member_le_sum
tff(fact_8901_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A3: set_VEBT_VEBT,Fa: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I),A3)
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),minus_5127226145743854075T_VEBT(A3,insert_VEBT_VEBT(I,bot_bo8194388402131092736T_VEBT)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_VEBT_VEBT_rat(Fa,X2)) )
       => ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(Fa,I)),groups136491112297645522BT_rat(Fa,A3)) ) ) ) ).

% member_le_sum
tff(fact_8902_member__le__sum,axiom,
    ! [I: complex,A3: set_complex,Fa: fun_complex_rat] :
      ( aa_set_complex_o(member_complex(I),A3)
     => ( ! [X2: complex] :
            ( aa_set_complex_o(member_complex(X2),minus_811609699411566653omplex(A3,insert_complex(I,bot_bot_set_complex)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(Fa,X2)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(Fa,I)),groups5058264527183730370ex_rat(Fa,A3)) ) ) ) ).

% member_le_sum
tff(fact_8903_member__le__sum,axiom,
    ! [I: code_integer,A3: set_Code_integer,Fa: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I),A3)
     => ( ! [X2: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X2),minus_2355218937544613996nteger(A3,insert_Code_integer(I,bot_bo3990330152332043303nteger)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Code_integer_rat(Fa,X2)) )
       => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(Fa,I)),groups6602215022474089585er_rat(Fa,A3)) ) ) ) ).

% member_le_sum
tff(fact_8904_member__le__sum,axiom,
    ! [I: real,A3: set_real,Fa: fun_real_rat] :
      ( aa_set_real_o(member_real(I),A3)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),minus_minus_set_real(A3,insert_real(I,bot_bot_set_real)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(Fa,X2)) )
       => ( aa_set_real_o(finite_finite_real,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(Fa,I)),groups1300246762558778688al_rat(Fa,A3)) ) ) ) ).

% member_le_sum
tff(fact_8905_member__le__sum,axiom,
    ! [I: int,A3: set_int,Fa: fun_int_rat] :
      ( aa_set_int_o(member_int(I),A3)
     => ( ! [X2: int] :
            ( aa_set_int_o(member_int(X2),minus_minus_set_int(A3,insert_int(I,bot_bot_set_int)))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Fa,X2)) )
       => ( aa_set_int_o(finite_finite_int,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(Fa,I)),groups3906332499630173760nt_rat(Fa,A3)) ) ) ) ).

% member_le_sum
tff(fact_8906_convex__sum__bound__le,axiom,
    ! [I5: set_VEBT_VEBT,Xb: fun_VEBT_VEBT_int,A2: fun_VEBT_VEBT_int,B2: int,Delta: int] :
      ( ! [I2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_VEBT_VEBT_int(Xb,I2)) )
     => ( ( groups769130701875090982BT_int(Xb,I5) = one_one_int )
       => ( ! [I2: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(I2),I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_VEBT_VEBT_int(A2,I2)),B2))),Delta) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,groups769130701875090982BT_int(aa_fun7144222995531728252BT_int(aTP_Lamm_any(Xb),A2),I5)),B2))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8907_convex__sum__bound__le,axiom,
    ! [I5: set_real,Xb: fun_real_int,A2: fun_real_int,B2: int,Delta: int] :
      ( ! [I2: real] :
          ( aa_set_real_o(member_real(I2),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(Xb,I2)) )
     => ( ( groups1932886352136224148al_int(Xb,I5) = one_one_int )
       => ( ! [I2: real] :
              ( aa_set_real_o(member_real(I2),I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(A2,I2)),B2))),Delta) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aTP_Lamm_anz(Xb),A2),I5)),B2))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8908_convex__sum__bound__le,axiom,
    ! [I5: set_nat,Xb: fun_nat_real,A2: fun_nat_real,B2: real,Delta: real] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Xb,I2)) )
     => ( ( groups6591440286371151544t_real(Xb,I5) = one_one_real )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(A2,I2)),B2))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_aoa(Xb),A2),I5)),B2))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8909_convex__sum__bound__le,axiom,
    ! [I5: set_int,Xb: fun_int_int,A2: fun_int_int,B2: int,Delta: int] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(Xb,I2)) )
     => ( ( groups4538972089207619220nt_int(Xb,I5) = one_one_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(A2,I2)),B2))),Delta) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_aob(Xb),A2),I5)),B2))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_8910_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_8911_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_8912_sin__arccos__nonzero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xb)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_8913_arccos__cos2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Xb)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xb)) = aa_real_real(uminus_uminus_real,Xb) ) ) ) ).

% arccos_cos2
tff(fact_8914_arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi)
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_8915_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fg,N))) ).

% mask_eq_sum_exp_nat
tff(fact_8916_gauss__sum__nat,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_eb),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(suc,N))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_8917_arith__series__nat,axiom,
    ! [A2: nat,D: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aoc(A2),D)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_8918_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_eb),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_8919_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) ) ) ).

% arccos_le_pi2
tff(fact_8920_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] : aa_real_real(arccos,aa_real_real(cos_real,Theta)) != aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Theta),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,K2)),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_8921_sin__arccos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xb)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_8922_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_8923_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,N) = bot_bot_set_nat )
    <=> ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_8924_lessThan__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K))) ).

% lessThan_nat_numeral
tff(fact_8925_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),N)
     => ( groups7754918857620584856omplex(aTP_Lamm_aod,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_acz(N),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_8926_sum__roots__unity,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),N)
     => ( groups7754918857620584856omplex(aTP_Lamm_aod,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_ach,N))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_8927_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_aoe),one_one_real) ).

% power_half_series
tff(fact_8928_sums__if_H,axiom,
    ! [G: fun_nat_real,Xb: real] :
      ( aa_real_o(sums_real(G),Xb)
     => aa_real_o(sums_real(aTP_Lamm_aof(G)),Xb) ) ).

% sums_if'
tff(fact_8929_sums__if,axiom,
    ! [G: fun_nat_real,Xb: real,Fa: fun_nat_real,Y: real] :
      ( aa_real_o(sums_real(G),Xb)
     => ( aa_real_o(sums_real(Fa),Y)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_aog(G),Fa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ) ) ).

% sums_if
tff(fact_8930_sum__split__even__odd,axiom,
    ! [Fa: fun_nat_real,G: fun_nat_real,N: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_aoh(Fa),G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_aoi(Fa),aa_nat_set_nat(set_ord_lessThan_nat,N))),groups6591440286371151544t_real(aTP_Lamm_aoj(G),aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% sum_split_even_odd
tff(fact_8931_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),N)
     => ( groups4538972089207619220nt_int(aTP_Lamm_ja,set_or1266510415728281911st_int(M,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,N),aa_int_int(aa_int_fun_int_int(plus_plus_int,N),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_8932_sum__pos__lt__pair,axiom,
    ! [Fa: fun_nat_real,K: nat] :
      ( summable_real(Fa)
     => ( ! [D2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Fa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D2)))),aa_nat_real(Fa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D2)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(Fa,aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(Fa)) ) ) ).

% sum_pos_lt_pair
tff(fact_8933_sumr__cos__zero__one,axiom,
    ! [N: nat] : groups6591440286371151544t_real(aTP_Lamm_aok,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_8934_sin__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_aol,Xb)),aa_real_real(sin_real,Xb)) ).

% sin_paired
tff(fact_8935_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_8936_square__fact__le__2__fact,axiom,
    ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,N)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% square_fact_le_2_fact
tff(fact_8937_cos__coeff__def,axiom,
    ! [X3: nat] :
      aa_nat_real(cos_coeff,X3) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X3)),zero_zero_real) ).

% cos_coeff_def
tff(fact_8938_Maclaurin__lemma,axiom,
    ! [H: real,Fa: fun_real_real,J: fun_nat_real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ? [B8: real] : aa_real_real(Fa,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_aom(H),J),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(B8),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N)),aa_nat_real(semiri2265585572941072030t_real,N)))) ) ).

% Maclaurin_lemma
tff(fact_8939_Maclaurin__cos__expansion,axiom,
    ! [Xb: real,N: nat] :
    ? [T: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aon,Xb),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_8940_Maclaurin__cos__expansion2,axiom,
    ! [Xb: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),Xb)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aon,Xb),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_8941_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),zero_zero_real)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aon,Xb),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_8942_cos__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_aoo,Xb)),aa_real_real(cos_real,Xb)) ).

% cos_paired
tff(fact_8943_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),Xb)
            & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aop,Xb),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_8944_Maclaurin__sin__expansion4,axiom,
    ! [Xb: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [T: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),Xb)
          & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aop,Xb),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_8945_Maclaurin__sin__expansion2,axiom,
    ! [Xb: real,N: nat] :
    ? [T: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aop,Xb),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_8946_Maclaurin__sin__expansion,axiom,
    ! [Xb: real,N: nat] :
    ? [T: real] : aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aop,Xb),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,N))),pi)))),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N))) ).

% Maclaurin_sin_expansion
tff(fact_8947_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_8948_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ) ).

% fact_mono_nat
tff(fact_8949_fact__ge__self,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ).

% fact_ge_self
tff(fact_8950_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ) ) ).

% fact_less_mono_nat
tff(fact_8951_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ).

% fact_ge_Suc_0_nat
tff(fact_8952_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ) ) ).

% dvd_fact
tff(fact_8953_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(suc,M))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N))) ) ) ).

% fact_diff_Suc
tff(fact_8954_fact__div__fact__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),R)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R)) ) ).

% fact_div_fact_le_pow
tff(fact_8955_sin__coeff__def,axiom,
    ! [X3: nat] :
      aa_nat_real(sin_coeff,X3) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3),zero_zero_real,aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X3))) ).

% sin_coeff_def
tff(fact_8956_Maclaurin__exp__lt,axiom,
    ! [Xb: real,N: nat] :
      ( ( Xb != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aoq,Xb),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(exp_real,T)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_8957_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,vEBT_Leaf((A2),(B2))) = zero_zero_nat ).

% VEBT_internal.height.simps(1)
tff(fact_8958_exp__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) ) ).

% exp_less_mono
tff(fact_8959_exp__less__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% exp_less_cancel_iff
tff(fact_8960_exp__eq__one__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(exp_real,Xb) = one_one_real )
    <=> ( Xb = zero_zero_real ) ) ).

% exp_eq_one_iff
tff(fact_8961_exp__less__one__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% exp_less_one_iff
tff(fact_8962_one__less__exp__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% one_less_exp_iff
tff(fact_8963_exp__le__one__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% exp_le_one_iff
tff(fact_8964_one__le__exp__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(exp_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% one_le_exp_iff
tff(fact_8965_exp__ln__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xb)) = Xb )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% exp_ln_iff
tff(fact_8966_exp__ln,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xb)) = Xb ) ) ).

% exp_ln
tff(fact_8967_exp__less__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% exp_less_cancel
tff(fact_8968_exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X2: real] : aa_real_real(exp_real,X2) = Y ) ).

% exp_total
tff(fact_8969_exp__gt__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(exp_real,Xb)) ).

% exp_gt_zero
tff(fact_8970_not__exp__less__zero,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),zero_zero_real) ).

% not_exp_less_zero
tff(fact_8971_not__exp__le__zero,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),zero_zero_real) ).

% not_exp_le_zero
tff(fact_8972_exp__ge__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,Xb)) ).

% exp_ge_zero
tff(fact_8973_exp__gt__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xb)) ) ).

% exp_gt_one
tff(fact_8974_exp__ge__add__one__self__aux,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(exp_real,Xb)) ) ).

% exp_ge_add_one_self_aux
tff(fact_8975_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y)
     => ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),aa_real_real(minus_minus_real(Y),one_one_real))
          & ( aa_real_real(exp_real,X2) = Y ) ) ) ).

% lemma_exp_total
tff(fact_8976_ln__ge__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Y)),Xb) ) ) ).

% ln_ge_iff
tff(fact_8977_exp__le,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),numeral_numeral_real(aa_num_num(bit1,one))) ).

% exp_le
tff(fact_8978_exp__half__le2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))),numeral_numeral_real(aa_num_num(bit0,one))) ).

% exp_half_le2
tff(fact_8979_VEBT__internal_Oheight_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A: $o,B: $o] : Xb != vEBT_Leaf((A),(B))
     => ~ ! [Uu2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xb != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg,TreeList2),Summary) ) ).

% VEBT_internal.height.cases
tff(fact_8980_exp__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% exp_bound
tff(fact_8981_real__exp__bound__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),Xb))) ) ) ).

% real_exp_bound_lemma
tff(fact_8982_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(divide_divide_real(Xb),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,Xb)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_8983_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xb: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(one_one_real),aa_real_real(divide_divide_real(Xb),aa_nat_real(semiri5074537144036343181t_real,N)))),N)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_8984_Maclaurin__exp__le,axiom,
    ! [Xb: real,N: nat] :
    ? [T: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aoq,Xb),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(exp_real,T)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N))) ) ) ).

% Maclaurin_exp_le
tff(fact_8985_exp__lower__Taylor__quadratic,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),numeral_numeral_real(aa_num_num(bit0,one))))),aa_real_real(exp_real,Xb)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_8986_log__base__10__eq2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xb) = aa_real_real(times_times_real(aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,Xb)) ) ) ).

% log_base_10_eq2
tff(fact_8987_tanh__real__altdef,axiom,
    ! [Xb: real] : aa_real_real(tanh_real,Xb) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_real_real(exp_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one)))),Xb)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(exp_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,numeral_numeral_real(aa_num_num(bit0,one)))),Xb)))) ).

% tanh_real_altdef
tff(fact_8988_log__base__10__eq1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xb) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,Xb)) ) ) ).

% log_base_10_eq1
tff(fact_8989_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_8990_floor__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(numeral_numeral_real(A2)),numeral_numeral_real(B2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A2)),aa_num_int(numeral_numeral_int,B2)) ).

% floor_divide_eq_div_numeral
tff(fact_8991_floor__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(B2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,B2)) ).

% floor_one_divide_eq_div_numeral
tff(fact_8992_floor__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(numeral_numeral_real(A2)),numeral_numeral_real(B2)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A2))),aa_num_int(numeral_numeral_int,B2)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_8993_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(B2)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,B2)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_8994_floor__eq,axiom,
    ! [N: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,N)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = N ) ) ) ).

% floor_eq
tff(fact_8995_real__of__int__floor__add__one__gt,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,R),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))),one_one_real)) ).

% real_of_int_floor_add_one_gt
tff(fact_8996_real__of__int__floor__gt__diff__one,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(R),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))) ).

% real_of_int_floor_gt_diff_one
tff(fact_8997_floor__eq2,axiom,
    ! [N: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,N)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,N)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = N ) ) ) ).

% floor_eq2
tff(fact_8998_floor__divide__real__eq__div,axiom,
    ! [B2: int,A2: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(A2),aa_int_real(ring_1_of_int_real,B2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_real_int(archim6058952711729229775r_real,A2)),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_8999_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),groups708209901874060359at_nat(aTP_Lamm_eb,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,N)),M))) ) ) ).

% fact_eq_fact_times
tff(fact_9000_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) = groups708209901874060359at_nat(aTP_Lamm_eb,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M)) ) ) ).

% fact_div_fact
tff(fact_9001_floor__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(B2),Xb)) = K )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(B2),aa_int_real(ring_1_of_int_real,K))),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(powr_real(B2),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_9002_floor__log2__div2,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_9003_floor__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_9004_Maclaurin__sin__bound,axiom,
    ! [Xb: real,N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(sin_real,Xb)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_aop,Xb),aa_nat_set_nat(set_ord_lessThan_nat,N))))),aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Xb)),N))) ).

% Maclaurin_sin_bound
tff(fact_9005__092_060open_062heap_Oadmissible_A_I_092_060lambda_062vebt__predi_H_O_A_092_060forall_062x_Axa_Axb_O_Arefines_A_Ivebt__predi_Axa_Axb_J_A_Icurry_A_Icurry_Avebt__predi_H_J_Ax_Axa_Axb_J_J_092_060close_062,axiom,
    comple5664458124825824345on_nat(partia6039416512482706817Ti_nat(heap_T7048022066654196708on_nat),partia7778075537949639097Ti_nat(heap_T7875578835903804844on_nat),aTP_Lamm_aor) ).

% \<open>heap.admissible (\<lambda>vebt_predi'. \<forall>x xa xb. refines (vebt_predi xa xb) (curry (curry vebt_predi') x xa xb))\<close>
tff(fact_9006_inverse__powr,axiom,
    ! [Y: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_real(powr_real(aa_real_real(inverse_inverse_real,Y)),A2) = aa_real_real(inverse_inverse_real,aa_real_real(powr_real(Y),A2)) ) ) ).

% inverse_powr
tff(fact_9007_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,I),J)) = groups1705073143266064639nt_int(aTP_Lamm_ja,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_9008_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E2: real] :
      ( ! [D2: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E)
         => ( aa_real_o(P,D2)
           => aa_real_o(P,E) ) )
     => ( ! [N3: nat] : aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N3))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(P,E2) ) ) ) ).

% forall_pos_mono_1
tff(fact_9009_forall__pos__mono,axiom,
    ! [P: fun_real_o,E2: real] :
      ( ! [D2: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E)
         => ( aa_real_o(P,D2)
           => aa_real_o(P,E) ) )
     => ( ! [N3: nat] :
            ( ( N3 != zero_zero_nat )
           => aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N3))) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(P,E2) ) ) ) ).

% forall_pos_mono
tff(fact_9010_real__arch__inverse,axiom,
    ! [E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
    <=> ? [N8: nat] :
          ( ( N8 != zero_zero_nat )
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N8)))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N8))),E2) ) ) ).

% real_arch_inverse
tff(fact_9011_sqrt__divide__self__eq,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(divide_divide_real(aa_real_real(sqrt,Xb)),Xb) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb)) ) ) ).

% sqrt_divide_self_eq
tff(fact_9012_ln__inverse,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_inverse
tff(fact_9013_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_ja,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)))) ).

% prod_int_plus_eq
tff(fact_9014_log__inverse,axiom,
    ! [A2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(log(A2),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(log(A2),Xb)) ) ) ) ) ).

% log_inverse
tff(fact_9015_exp__plus__inverse__exp,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))) ).

% exp_plus_inverse_exp
tff(fact_9016_plus__inverse__ge__2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,numeral_numeral_real(aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(inverse_inverse_real,Xb))) ) ).

% plus_inverse_ge_2
tff(fact_9017_real__inv__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,Xb) ) ) ).

% real_inv_sqrt_pow2
tff(fact_9018_tan__cot,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))),Xb)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,Xb)) ).

% tan_cot
tff(fact_9019_real__le__x__sinh,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))),numeral_numeral_real(aa_num_num(bit0,one)))) ) ).

% real_le_x_sinh
tff(fact_9020_real__le__abs__sinh,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))),numeral_numeral_real(aa_num_num(bit0,one))))) ).

% real_le_abs_sinh
tff(fact_9021_binomial__code,axiom,
    ! [N: nat,K: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),one_one_nat)),N),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ).

% binomial_code
tff(fact_9022_binomial__n__n,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),N) = one_one_nat ).

% binomial_n_n
tff(fact_9023_binomial__0__Suc,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_9024_binomial__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,zero_zero_nat)) = N ).

% binomial_1
tff(fact_9025_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K) ) ).

% binomial_eq_0_iff
tff(fact_9026_binomial__n__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_9027_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) ) ).

% zero_less_binomial_iff
tff(fact_9028_atMost__0,axiom,
    aa_nat_set_nat(set_ord_atMost_nat,zero_zero_nat) = insert_nat(zero_zero_nat,bot_bot_set_nat) ).

% atMost_0
tff(fact_9029_sum__choose__upper,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aos,M)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,M)) ).

% sum_choose_upper
tff(fact_9030_choose__one,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),one_one_nat) = N ).

% choose_one
tff(fact_9031_sum__choose__lower,axiom,
    ! [R: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aot,R)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R),N))),N) ).

% sum_choose_lower
tff(fact_9032_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aou,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_9033_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aou,N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_9034_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_9035_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)) ) ) ).

% binomial_symmetric
tff(fact_9036_binomial__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),R)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,N),R)) ) ).

% binomial_le_pow
tff(fact_9037_atMost__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N) ).

% atMost_atLeast0
tff(fact_9038_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aov(M),N)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_9039_vandermonde,axiom,
    ! [M: nat,N: nat,R: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_aow(M),N),R)),aa_nat_set_nat(set_ord_atMost_nat,R)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),R) ).

% vandermonde
tff(fact_9040_choose__row__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% choose_row_sum
tff(fact_9041_binomial,axiom,
    ! [A2: nat,B2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),N) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_aox(A2),B2),N)),aa_nat_set_nat(set_ord_atMost_nat,N)) ).

% binomial
tff(fact_9042_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) ) ).

% zero_less_binomial
tff(fact_9043_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,N)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,N)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_9044_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),M)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_9045_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_9046_choose__square__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aoy,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N) ).

% choose_square_sum
tff(fact_9047_atMost__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(numeral_numeral_nat,K),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K))) ).

% atMost_nat_numeral
tff(fact_9048_binomial__absorption,axiom,
    ! [K: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_9049_binomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% binomial_r_part_sum
tff(fact_9050_choose__linear__sum,axiom,
    ! [N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_aoz,N)),aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).

% choose_linear_sum
tff(fact_9051_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_9052_binomial__maximum,axiom,
    ! [N: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% binomial_maximum
tff(fact_9053_binomial__antimono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K4)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) ) ) ) ).

% binomial_antimono
tff(fact_9054_binomial__mono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K4)),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K4)) ) ) ).

% binomial_mono
tff(fact_9055_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N)) ).

% binomial_maximum'
tff(fact_9056_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% binomial_le_pow2
tff(fact_9057_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_9058_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_9059_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_9060_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_9061_binomial__strict__mono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K4)),N)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K4)) ) ) ).

% binomial_strict_mono
tff(fact_9062_binomial__strict__antimono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K4)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_9063_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_9064_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_9065_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,aa_nat_nat(suc,zero_zero_nat)),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9066_choose__two,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_9067_polynomial__product__nat,axiom,
    ! [M: nat,A2: fun_nat_nat,N: nat,B2: fun_nat_nat,Xb: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I2)
         => ( aa_nat_nat(A2,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J2)
           => ( aa_nat_nat(B2,J2) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_apa(A2),Xb)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_apa(B2),Xb)),aa_nat_set_nat(set_ord_atMost_nat,N))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_apc(A2),B2),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_9068_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,numeral_numeral_real(aa_num_num(bit0,aa_num_num(bit0,one)))),N)),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),N))) ) ).

% central_binomial_lower_bound
tff(fact_9069_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T)),aa_real_real(sin_real,T)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_9070_sinh__real__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sinh_real,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% sinh_real_zero_iff
tff(fact_9071_sinh__real__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xb)),aa_real_real(sinh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% sinh_real_less_iff
tff(fact_9072_of__nat__id,axiom,
    ! [N: nat] : aa_nat_nat(semiri1316708129612266289at_nat,N) = N ).

% of_nat_id
tff(fact_9073_sinh__real__neg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% sinh_real_neg_iff
tff(fact_9074_sinh__real__pos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sinh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% sinh_real_pos_iff
tff(fact_9075_sinh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% sinh_real_nonneg_iff
tff(fact_9076_sinh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% sinh_real_nonpos_iff
tff(fact_9077_sinh__less__cosh__real,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Xb)) ).

% sinh_less_cosh_real
tff(fact_9078_cosh__real__nonzero,axiom,
    ! [Xb: real] : aa_real_real(cosh_real,Xb) != zero_zero_real ).

% cosh_real_nonzero
tff(fact_9079_Complex__eq__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( aa_real_complex(complex2(A2),B2) = numera6690914467698888265omplex(W) )
    <=> ( ( A2 = numeral_numeral_real(W) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_9080_cosh__real__pos,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cosh_real,Xb)) ).

% cosh_real_pos
tff(fact_9081_cosh__real__nonpos__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_9082_cosh__real__nonneg__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_9083_cosh__real__nonneg,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,Xb)) ).

% cosh_real_nonneg
tff(fact_9084_complex__eq__cancel__iff2,axiom,
    ! [Xb: real,Y: real,Xaa: real] :
      ( ( aa_real_complex(complex2(Xb),Y) = aa_real_complex(real_V4546457046886955230omplex,Xaa) )
    <=> ( ( Xb = Xaa )
        & ( Y = zero_zero_real ) ) ) ).

% complex_eq_cancel_iff2
tff(fact_9085_complex__of__real__code,axiom,
    ! [X3: real] : aa_real_complex(real_V4546457046886955230omplex,X3) = aa_real_complex(complex2(X3),zero_zero_real) ).

% complex_of_real_code
tff(fact_9086_complex__of__real__def,axiom,
    ! [R: real] : aa_real_complex(real_V4546457046886955230omplex,R) = aa_real_complex(complex2(R),zero_zero_real) ).

% complex_of_real_def
tff(fact_9087_zero__complex_Ocode,axiom,
    zero_zero_complex = aa_real_complex(complex2(zero_zero_real),zero_zero_real) ).

% zero_complex.code
tff(fact_9088_Complex__eq__0,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_complex(complex2(A2),B2) = zero_zero_complex )
    <=> ( ( A2 = zero_zero_real )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_0
tff(fact_9089_cosh__real__strict__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_9090_cosh__real__nonneg__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_9091_cosh__real__nonpos__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_9092_Complex__eq__neg__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( aa_real_complex(complex2(A2),B2) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> ( ( A2 = aa_real_real(uminus_uminus_real,numeral_numeral_real(W)) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_9093_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_9094_Complex__eq__1,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_complex(complex2(A2),B2) = one_one_complex )
    <=> ( ( A2 = one_one_real )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_1
tff(fact_9095_arcosh__cosh__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,Xb)) = Xb ) ) ).

% arcosh_cosh_real
tff(fact_9096_Complex__eq__neg__1,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_complex(complex2(A2),B2) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( A2 = aa_real_real(uminus_uminus_real,one_one_real) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
tff(fact_9097_complex__norm,axiom,
    ! [Xb: real,Y: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(Xb),Y)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_norm
tff(fact_9098_complex__inverse,axiom,
    ! [A2: real,B2: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(complex2(A2),B2)) = aa_real_complex(complex2(aa_real_real(divide_divide_real(A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_9099_cosh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(inverse_inverse_real,Xb))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_9100_sinh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Xb),aa_real_real(inverse_inverse_real,Xb))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_9101_cot__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,Xb)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_9102_cot__pi,axiom,
    aa_real_real(cot_real,pi) = zero_zero_real ).

% cot_pi
tff(fact_9103_cot__npi,axiom,
    ! [N: nat] : aa_real_real(cot_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_9104_cot__periodic,axiom,
    ! [Xb: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,Xb) ).

% cot_periodic
tff(fact_9105_cot__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cot_real,Xb)) ) ) ).

% cot_gt_zero
tff(fact_9106_tan__cot_H,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))),Xb)) = aa_real_real(cot_real,Xb) ).

% tan_cot'
tff(fact_9107_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique5052692396658037445od_int(aa_num_num(bitM,M),aa_num_num(bit0,one)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_9108_i__even__power,axiom,
    ! [N: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),N) ).

% i_even_power
tff(fact_9109_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,K)) ).

% pred_numeral_simps(2)
tff(fact_9110_power2__i,axiom,
    aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,imaginary_unit),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power2_i
tff(fact_9111_exp__two__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_real_complex(real_V4546457046886955230omplex,pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_9112_exp__two__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_9113_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero_complex ).

% complex_i_not_zero
tff(fact_9114_semiring__norm_I26_J,axiom,
    aa_num_num(bitM,one) = one ).

% semiring_norm(26)
tff(fact_9115_semiring__norm_I27_J,axiom,
    ! [N: num] : aa_num_num(bitM,aa_num_num(bit0,N)) = aa_num_num(bit1,aa_num_num(bitM,N)) ).

% semiring_norm(27)
tff(fact_9116_semiring__norm_I28_J,axiom,
    ! [N: num] : aa_num_num(bitM,aa_num_num(bit1,N)) = aa_num_num(bit1,aa_num_num(bit0,N)) ).

% semiring_norm(28)
tff(fact_9117_eval__nat__numeral_I2_J,axiom,
    ! [N: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,N)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,N))) ).

% eval_nat_numeral(2)
tff(fact_9118_one__plus__BitM,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bitM,N)) = aa_num_num(bit0,N) ).

% one_plus_BitM
tff(fact_9119_BitM__plus__one,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bitM,N)),one) = aa_num_num(bit0,N) ).

% BitM_plus_one
tff(fact_9120_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_9121_Complex__eq__i,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_complex(complex2(Xb),Y) = imaginary_unit )
    <=> ( ( Xb = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_9122_complex__of__real__i,axiom,
    ! [R: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,R)),imaginary_unit) = aa_real_complex(complex2(zero_zero_real),R) ).

% complex_of_real_i
tff(fact_9123_i__complex__of__real,axiom,
    ! [R: real] : aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,R)) = aa_real_complex(complex2(zero_zero_real),R) ).

% i_complex_of_real
tff(fact_9124_Arg__minus__ii,axiom,
    aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),numeral_numeral_real(aa_num_num(bit0,one))) ).

% Arg_minus_ii
tff(fact_9125_csqrt__ii,axiom,
    aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,one_one_complex),imaginary_unit)),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,numeral_numeral_real(aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_9126_Arg__ii,axiom,
    aa_complex_real(arg,imaginary_unit) = aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))) ).

% Arg_ii
tff(fact_9127_csqrt__eq__0,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(csqrt,Z) = zero_zero_complex )
    <=> ( Z = zero_zero_complex ) ) ).

% csqrt_eq_0
tff(fact_9128_csqrt__0,axiom,
    aa_complex_complex(csqrt,zero_zero_complex) = zero_zero_complex ).

% csqrt_0
tff(fact_9129_power2__csqrt,axiom,
    ! [Z: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(csqrt,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z ).

% power2_csqrt
tff(fact_9130_Arg__zero,axiom,
    aa_complex_real(arg,zero_zero_complex) = zero_zero_real ).

% Arg_zero
tff(fact_9131_of__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,Xb)) = aa_complex_complex(csqrt,aa_real_complex(real_V4546457046886955230omplex,Xb)) ) ) ).

% of_real_sqrt
tff(fact_9132_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ).

% Arg_bounded
tff(fact_9133_cis__minus__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_9134_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_apd(D))) ).

% int_ge_less_than_def
tff(fact_9135_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_ape(D))) ).

% int_ge_less_than2_def
tff(fact_9136_cis__zero,axiom,
    aa_real_complex(cis,zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_9137_cis__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_9138_cis__2pi,axiom,
    aa_real_complex(cis,aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_9139_cis__neq__zero,axiom,
    ! [A2: real] : aa_real_complex(cis,A2) != zero_zero_complex ).

% cis_neq_zero
tff(fact_9140_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_set_complex_o(bij_betw_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_apf,N),aa_nat_set_nat(set_ord_lessThan_nat,N)),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_ach,N))) ) ).

% bij_betw_roots_unity
tff(fact_9141_set__decode__0,axiom,
    ! [Xb: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,Xb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb) ) ).

% set_decode_0
tff(fact_9142_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_9143_set__decode__Suc,axiom,
    ! [N: nat,Xb: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,N)),aa_nat_set_nat(nat_set_decode,Xb))
    <=> aa_set_nat_o(member_nat(N),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% set_decode_Suc
tff(fact_9144_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(nat_set_decode,M)),aa_nat_set_nat(nat_set_decode,N))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N) ) ).

% subset_decode_imp_le
tff(fact_9145_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ aa_set_nat_o(member_nat(N),aa_nat_set_nat(nat_set_decode,Z))
     => ( aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),Z)) = insert_nat(N,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_9146_set__decode__def,axiom,
    ! [Xb: nat] : aa_nat_set_nat(nat_set_decode,Xb) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_apg,Xb)) ).

% set_decode_def
tff(fact_9147_forall__finite_I3_J,axiom,
    ! [Xb: nat,P: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,aa_nat_nat(suc,Xb)))
         => aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Xb))
           => aa_nat_o(P,aa_nat_nat(suc,I4)) ) ) ) ).

% forall_finite(3)
tff(fact_9148_forall__finite_I2_J,axiom,
    ! [P: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,zero_zero_nat))
         => aa_nat_o(P,I4) )
    <=> aa_nat_o(P,zero_zero_nat) ) ).

% forall_finite(2)
tff(fact_9149_Suc__0__mod__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_o_nat(zero_n2687167440665602831ol_nat,N != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_9150_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R: int] : aa_Pro6799258434465214121nt_int(adjust_div,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q2),aa_o_int(zero_n2684676970156552555ol_int,R != zero_zero_int)) ).

% Divides.adjust_div_eq
tff(fact_9151_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : aa_Pro6799258434465214121nt_int(adjust_div,Qr) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aTP_Lamm_aph),Qr) ).

% Divides.adjust_div_def
tff(fact_9152_forall__finite_I1_J,axiom,
    ! [P: fun_nat_o,I3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),zero_zero_nat)
     => aa_nat_o(P,I3) ) ).

% forall_finite(1)
tff(fact_9153_Comparator__Generator_OAll__less__Suc,axiom,
    ! [Xb: nat,P: fun_nat_o] :
      ( ! [I4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(suc,Xb))
         => aa_nat_o(P,I4) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),Xb)
           => aa_nat_o(P,aa_nat_nat(suc,I4)) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_9154_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_set_complex_o(bij_be1856998921033663316omplex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,N),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(divide_divide_real(aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,N))))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_ach,N))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_api(C),N))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9155_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
        ( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
        aa_int_int(uminus_uminus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,
            aa_o_int(zero_n2684676970156552555ol_int,
              ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
              & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ).

% and_int.simps
tff(fact_9156_and__int_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Xaa) = Y )
     => ( Y = $ite(
            ( aa_set_int_o(member_int(Xb),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int(Xaa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.elims
tff(fact_9157_real__root__zero,axiom,
    ! [N: nat] : aa_real_real(aa_nat_fun_real_real(root,N),zero_zero_real) = zero_zero_real ).

% real_root_zero
tff(fact_9158_real__root__Suc__0,axiom,
    ! [Xb: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),Xb) = Xb ).

% real_root_Suc_0
tff(fact_9159_real__root__eq__iff,axiom,
    ! [N: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),Xb) = aa_real_real(aa_nat_fun_real_real(root,N),Y) )
      <=> ( Xb = Y ) ) ) ).

% real_root_eq_iff
tff(fact_9160_root__0,axiom,
    ! [Xb: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),Xb) = zero_zero_real ).

% root_0
tff(fact_9161_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_9162_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% and_negative_int_iff
tff(fact_9163_real__root__eq__0__iff,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),Xb) = zero_zero_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9164_real__root__less__iff,axiom,
    ! [N: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),aa_real_real(aa_nat_fun_real_real(root,N),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% real_root_less_iff
tff(fact_9165_real__root__le__iff,axiom,
    ! [N: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),aa_real_real(aa_nat_fun_real_real(root,N),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% real_root_le_iff
tff(fact_9166_real__root__one,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_real(aa_nat_fun_real_real(root,N),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9167_real__root__eq__1__iff,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,N),Xb) = one_one_real )
      <=> ( Xb = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9168_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_9169_real__root__lt__0__iff,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_9170_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_9171_real__root__le__0__iff,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_9172_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_9173_real__root__lt__1__iff,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_9174_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,N),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_9175_real__root__le__1__iff,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_9176_and__minus__numerals_I6_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_9177_and__minus__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_9178_real__root__pow__pos2,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),N) = Xb ) ) ) ).

% real_root_pow_pos2
tff(fact_9179_and__minus__numerals_I1_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_9180_and__minus__numerals_I5_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_9181_real__root__pos__pos__le,axiom,
    ! [Xb: real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Xb)) ) ).

% real_root_pos_pos_le
tff(fact_9182_AND__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)) ) ).

% AND_lower
tff(fact_9183_AND__upper1,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)),Xb) ) ).

% AND_upper1
tff(fact_9184_AND__upper2,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)),Y) ) ).

% AND_upper2
tff(fact_9185_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_9186_AND__upper2_H,axiom,
    ! [Y: int,Z: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_9187_real__root__less__mono,axiom,
    ! [N: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),aa_real_real(aa_nat_fun_real_real(root,N),Y)) ) ) ).

% real_root_less_mono
tff(fact_9188_real__root__le__mono,axiom,
    ! [N: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),aa_real_real(aa_nat_fun_real_real(root,N),Y)) ) ) ).

% real_root_le_mono
tff(fact_9189_real__root__power,axiom,
    ! [N: nat,Xb: real,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),K) ) ) ).

% real_root_power
tff(fact_9190_real__root__abs,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(abs_abs_real,Xb)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)) ) ) ).

% real_root_abs
tff(fact_9191_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),K) ) ).

% and_less_eq
tff(fact_9192_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_9193_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_9194_real__root__gt__zero,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Xb)) ) ) ).

% real_root_gt_zero
tff(fact_9195_real__root__strict__decreasing,axiom,
    ! [N: nat,N2: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N2),Xb)),aa_real_real(aa_nat_fun_real_real(root,N),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9196_sqrt__def,axiom,
    sqrt = aa_nat_fun_real_real(root,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% sqrt_def
tff(fact_9197_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_9198_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ) ) ).

% even_and_iff_int
tff(fact_9199_real__root__pos__pos,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,N),Xb)) ) ) ).

% real_root_pos_pos
tff(fact_9200_real__root__strict__increasing,axiom,
    ! [N: nat,N2: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),aa_real_real(aa_nat_fun_real_real(root,N2),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9201_real__root__decreasing,axiom,
    ! [N: nat,N2: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N2),Xb)),aa_real_real(aa_nat_fun_real_real(root,N),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_9202_odd__real__root__pow,axiom,
    ! [N: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),N) = Xb ) ) ).

% odd_real_root_pow
tff(fact_9203_odd__real__root__unique,axiom,
    ! [N: nat,Y: real,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N) = Xb )
       => ( aa_real_real(aa_nat_fun_real_real(root,N),Xb) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_9204_odd__real__root__power__cancel,axiom,
    ! [N: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)) = Xb ) ) ).

% odd_real_root_power_cancel
tff(fact_9205_real__root__pow__pos,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),N) = Xb ) ) ) ).

% real_root_pow_pos
tff(fact_9206_real__root__power__cancel,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)) = Xb ) ) ) ).

% real_root_power_cancel
tff(fact_9207_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),N) = Xb )
         => ( aa_real_real(aa_nat_fun_real_real(root,N),Xb) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_9208_real__root__increasing,axiom,
    ! [N: nat,N2: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),aa_real_real(aa_nat_fun_real_real(root,N2),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_9209_log__root,axiom,
    ! [N: nat,A2: real,B2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
       => ( aa_real_real(log(B2),aa_real_real(aa_nat_fun_real_real(root,N),A2)) = aa_real_real(divide_divide_real(aa_real_real(log(B2),A2)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% log_root
tff(fact_9210_log__base__root,axiom,
    ! [N: nat,B2: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => ( aa_real_real(log(aa_real_real(aa_nat_fun_real_real(root,N),B2)),Xb) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(log(B2),Xb)) ) ) ) ).

% log_base_root
tff(fact_9211_ln__root,axiom,
    ! [N: nat,B2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,N),B2)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B2)),aa_nat_real(semiri5074537144036343181t_real,N)) ) ) ) ).

% ln_root
tff(fact_9212_and__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_9213_root__powr__inverse,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,N),Xb) = aa_real_real(powr_real(Xb),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,N))) ) ) ) ).

% root_powr_inverse
tff(fact_9214_and__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
        ( ( K = zero_zero_int )
        | ( L = zero_zero_int ) ),
        zero_zero_int,
        $ite(
          K = aa_int_int(uminus_uminus_int,one_one_int),
          L,
          $ite(L = aa_int_int(uminus_uminus_int,one_one_int),K,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% and_int_unfold
tff(fact_9215_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K),L))
     => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
            ( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.psimps
tff(fact_9216_and__int_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Xaa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
       => ~ ( ( Y = $ite(
                  ( aa_set_int_o(member_int(Xb),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & aa_set_int_o(member_int(Xaa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
                  aa_int_int(uminus_uminus_int,
                    aa_o_int(zero_n2684676970156552555ol_int,
                      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
                  aa_int_int(aa_int_fun_int_int(plus_plus_int,
                      aa_o_int(zero_n2684676970156552555ol_int,
                        ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_9217_and__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_9218_and__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_9219_and__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_9220_and__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_9221_and__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_9222_Suc__0__and__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_9223_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = $ite(
        ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_9224_and__nat__rec,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_9225_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [K2: int,L2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K2),L2))
           => ( ( ~ ( aa_set_int_o(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & aa_set_int_o(member_int(L2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K2),L2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_9226_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [I2: int,J2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I2),J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(P,I2),J2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_9227_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : aa_num_option_num(bit_take_bit_num(zero_zero_nat),M) = none_num ).

% take_bit_num_simps(1)
tff(fact_9228_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] : aa_num_option_num(bit_take_bit_num(aa_nat_nat(suc,N)),one) = aa_num_option_num(some_num,one) ).

% take_bit_num_simps(2)
tff(fact_9229_take__bit__num__simps_I5_J,axiom,
    ! [R: num] : aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R)),one) = aa_num_option_num(some_num,one) ).

% take_bit_num_simps(5)
tff(fact_9230_pred__numeral__inc,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(inc,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% pred_numeral_inc
tff(fact_9231_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : aa_num_option_num(bit_take_bit_num(aa_nat_nat(suc,N)),aa_num_num(bit0,M)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_apj),aa_num_option_num(bit_take_bit_num(N),M)) ).

% take_bit_num_simps(3)
tff(fact_9232_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] : aa_num_option_num(bit_take_bit_num(aa_nat_nat(suc,N)),aa_num_num(bit1,M)) = aa_num_option_num(some_num,aa_option_num_num(case_option_num_num(one,bit1),aa_num_option_num(bit_take_bit_num(N),M))) ).

% take_bit_num_simps(4)
tff(fact_9233_take__bit__num__simps_I6_J,axiom,
    ! [R: num,M: num] : aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R)),aa_num_num(bit0,M)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_apj),aa_num_option_num(bit_take_bit_num(aa_num_nat(pred_numeral,R)),M)) ).

% take_bit_num_simps(6)
tff(fact_9234_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_Suc_0
tff(fact_9235_take__bit__num__simps_I7_J,axiom,
    ! [R: num,M: num] : aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R)),aa_num_num(bit1,M)) = aa_num_option_num(some_num,aa_option_num_num(case_option_num_num(one,bit1),aa_num_option_num(bit_take_bit_num(aa_num_nat(pred_numeral,R)),M))) ).

% take_bit_num_simps(7)
tff(fact_9236_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_option_num_int(case_option_int_num(zero_zero_int,aTP_Lamm_apk(M)),aa_num_option_num(bit_take_bit_num(aa_num_nat(numeral_numeral_nat,M)),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_9237_take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(bit_se2923211474154528505it_int(N),L))) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).

% take_bit_mult
tff(fact_9238_take__bit__minus,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,aa_int_int(bit_se2923211474154528505it_int(N),K))) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,K)) ).

% take_bit_minus
tff(fact_9239_take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(bit_se2923211474154528505it_int(N),L))) = aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% take_bit_diff
tff(fact_9240_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_9241_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_9242_num__induct,axiom,
    ! [P: fun_num_o,Xb: num] :
      ( aa_num_o(P,one)
     => ( ! [X2: num] :
            ( aa_num_o(P,X2)
           => aa_num_o(P,aa_num_num(inc,X2)) )
       => aa_num_o(P,Xb) ) ) ).

% num_induct
tff(fact_9243_add__inc,axiom,
    ! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),aa_num_num(inc,Y)) = aa_num_num(inc,aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),Y)) ).

% add_inc
tff(fact_9244_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(M),K)),aa_int_int(bit_se2923211474154528505it_int(N),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_9245_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% take_bit_nonnegative
tff(fact_9246_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_9247_not__take__bit__negative,axiom,
    ! [N: nat,K: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_9248_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_se2923211474154528505it_int(N),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_9249_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_9250_inc_Osimps_I3_J,axiom,
    ! [Xb: num] : aa_num_num(inc,aa_num_num(bit1,Xb)) = aa_num_num(bit0,aa_num_num(inc,Xb)) ).

% inc.simps(3)
tff(fact_9251_inc_Osimps_I2_J,axiom,
    ! [Xb: num] : aa_num_num(inc,aa_num_num(bit0,Xb)) = aa_num_num(bit1,Xb) ).

% inc.simps(2)
tff(fact_9252_add__One,axiom,
    ! [Xb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),one) = aa_num_num(inc,Xb) ).

% add_One
tff(fact_9253_inc__BitM__eq,axiom,
    ! [N: num] : aa_num_num(inc,aa_num_num(bitM,N)) = aa_num_num(bit0,N) ).

% inc_BitM_eq
tff(fact_9254_BitM__inc__eq,axiom,
    ! [N: num] : aa_num_num(bitM,aa_num_num(inc,N)) = aa_num_num(bit1,N) ).

% BitM_inc_eq
tff(fact_9255_mult__inc,axiom,
    ! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(times_times_num,Xb),aa_num_num(inc,Y)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(times_times_num,Xb),Y)),Xb) ).

% mult_inc
tff(fact_9256_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) != zero_zero_int )
     => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_9257_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_9258_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% take_bit_nat_less_exp
tff(fact_9259_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = M )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_9260_take__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% take_bit_nat_def
tff(fact_9261_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_9262_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% take_bit_int_less_exp
tff(fact_9263_take__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% take_bit_int_def
tff(fact_9264_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_9265_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),M)),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_9266_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_9267_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_se2923211474154528505it_int(N),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_9268_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_9269_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_9270_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_9271_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) )
     => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(bit_se2923211474154528505it_int(N),K)) ) ) ).

% take_bit_incr_eq
tff(fact_9272_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_numeral_minus_bit0
tff(fact_9273_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ) ).

% take_bit_int_less_eq
tff(fact_9274_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))),aa_int_int(bit_se2923211474154528505it_int(N),K)) ) ).

% take_bit_int_greater_eq
tff(fact_9275_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_9276_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,K)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_9277_uint32_Osize__eq,axiom,
    ! [P2: uint32] : size_size_uint32(P2) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))) ).

% uint32.size_eq
tff(fact_9278_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),aa_num_int(numeral_numeral_int,M)) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),aa_num_num(bitM,N))) ).

% and_minus_numerals(7)
tff(fact_9279_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),aa_num_num(bitM,N))) ).

% and_minus_numerals(3)
tff(fact_9280_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),aa_num_num(bit0,N))) ).

% and_minus_numerals(4)
tff(fact_9281_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,N))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ).

% mask_nat_positive_iff
tff(fact_9282_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),aa_num_int(numeral_numeral_int,M)) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),aa_num_num(bit0,N))) ).

% and_minus_numerals(8)
tff(fact_9283_less__eq__mask,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N)) ).

% less_eq_mask
tff(fact_9284_mask__nonnegative__int,axiom,
    ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,N)) ).

% mask_nonnegative_int
tff(fact_9285_not__mask__negative__int,axiom,
    ! [N: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_9286_and__not__num_Osimps_I1_J,axiom,
    aa_num_option_num(bit_and_not_num(one),one) = none_num ).

% and_not_num.simps(1)
tff(fact_9287_less__mask,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),N)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(bit_se2002935070580805687sk_nat,N)) ) ).

% less_mask
tff(fact_9288_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] : aa_num_option_num(bit_and_not_num(aa_num_num(bit0,M)),one) = aa_num_option_num(some_num,aa_num_num(bit0,M)) ).

% and_not_num.simps(4)
tff(fact_9289_and__not__num_Osimps_I2_J,axiom,
    ! [N: num] : aa_num_option_num(bit_and_not_num(one),aa_num_num(bit0,N)) = aa_num_option_num(some_num,one) ).

% and_not_num.simps(2)
tff(fact_9290_and__not__num_Osimps_I3_J,axiom,
    ! [N: num] : aa_num_option_num(bit_and_not_num(one),aa_num_num(bit1,N)) = none_num ).

% and_not_num.simps(3)
tff(fact_9291_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_9292_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] : aa_num_option_num(bit_and_not_num(aa_num_num(bit1,M)),one) = aa_num_option_num(some_num,aa_num_num(bit0,M)) ).

% and_not_num.simps(7)
tff(fact_9293_Suc__mask__eq__exp,axiom,
    ! [N: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% Suc_mask_eq_exp
tff(fact_9294_mask__nat__less__exp,axiom,
    ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% mask_nat_less_exp
tff(fact_9295_mask__half__int,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ).

% mask_half_int
tff(fact_9296_mask__int__def,axiom,
    ! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,N) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),one_one_int) ).

% mask_int_def
tff(fact_9297_mask__nat__def,axiom,
    ! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat) ).

% mask_nat_def
tff(fact_9298_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : aa_num_option_num(bit_and_not_num(aa_num_num(bit1,M)),aa_num_num(bit0,N)) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(aa_num_option_num(some_num,one),aTP_Lamm_apl),aa_num_option_num(bit_and_not_num(M),N)) ).

% and_not_num.simps(8)
tff(fact_9299_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_nat_int(bit_se2000444600071755411sk_int,N) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_9300_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M: num] : aa_num_option_num(bit_take_bit_num(N),aa_num_num(bit1,M)) = aa_nat_option_num(case_nat_option_num(none_num,aTP_Lamm_apm(M)),N) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_9301_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] : aa_num_option_num(bit_take_bit_num(N),one) = aa_nat_option_num(case_nat_option_num(none_num,aTP_Lamm_apn),N) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_9302_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] : aa_num_option_num(bit_take_bit_num(N),aa_num_num(bit0,M)) = aa_nat_option_num(case_nat_option_num(none_num,aTP_Lamm_apo(M)),N) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_9303_Bit__Operations_Otake__bit__num__code,axiom,
    ! [N: nat,M: num] : aa_num_option_num(bit_take_bit_num(N),M) = produc478579273971653890on_num(aTP_Lamm_aps,aa_num5026351239867140595at_num(product_Pair_nat_num(N),M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_9304_powr__int,axiom,
    ! [Xb: real,I: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),aa_int_real(ring_1_of_int_real,I)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,I)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I))))) ) ) ).

% powr_int
tff(fact_9305_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_apt,Y)) ).

% arctan_def
tff(fact_9306_nat__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_numeral
tff(fact_9307_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_9308_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int) ) ).

% nat_0_iff
tff(fact_9309_nat__le__0,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_9310_nat__neg__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_9311_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% zless_nat_conj
tff(fact_9312_nat__zminus__int,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_9313_int__nat__eq,axiom,
    ! [Z: int] :
      aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),Z,zero_zero_int) ).

% int_nat_eq
tff(fact_9314_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% zero_less_nat_eq
tff(fact_9315_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V3))) ).

% diff_nat_numeral
tff(fact_9316_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,N: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_9317_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_9318_nat__ceiling__le__eq,axiom,
    ! [Xb: real,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xb))),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,A2)) ) ).

% nat_ceiling_le_eq
tff(fact_9319_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% one_less_nat_eq
tff(fact_9320_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_9321_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_9322_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N)),aa_int_nat(nat2,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)),A2) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_9323_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_9324_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xb: num,N: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),N)),aa_int_nat(nat2,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),N)),A2) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_9325_nat__numeral__as__int,axiom,
    ! [X3: num] : aa_num_nat(numeral_numeral_nat,X3) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X3)) ).

% nat_numeral_as_int
tff(fact_9326_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_9327_nat__mono,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ).

% nat_mono
tff(fact_9328_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_9329_ex__nat,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
    <=> ? [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
          & aa_nat_o(P,aa_int_nat(nat2,X)) ) ) ).

% ex_nat
tff(fact_9330_all__nat,axiom,
    ! [P: fun_nat_o] :
      ( ! [X_12: nat] : aa_nat_o(P,X_12)
    <=> ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
         => aa_nat_o(P,aa_int_nat(nat2,X)) ) ) ).

% all_nat
tff(fact_9331_eq__nat__nat__iff,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z5) )
        <=> ( Z = Z5 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_9332_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_fn),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_9333_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_fs),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_9334_unset__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),N) = aa_int_nat(nat2,aa_int_int(bit_se4203085406695923979it_int(M),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% unset_bit_nat_def
tff(fact_9335_nat__mask__eq,axiom,
    ! [N: nat] : aa_int_nat(nat2,aa_nat_int(bit_se2000444600071755411sk_int,N)) = aa_nat_nat(bit_se2002935070580805687sk_nat,N) ).

% nat_mask_eq
tff(fact_9336_ln__real__def,axiom,
    ! [Xb: real] : aa_real_real(ln_ln_real,Xb) = the_real(aa_real_fun_real_o(aTP_Lamm_apu,Xb)) ).

% ln_real_def
tff(fact_9337_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_mono_iff
tff(fact_9338_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_9339_nat__le__iff,axiom,
    ! [Xb: int,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),N)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% nat_le_iff
tff(fact_9340_nat__0__le,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_9341_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
    <=> ( ( M = aa_int_nat(nat2,Z) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ) ).

% int_eq_iff
tff(fact_9342_nat__plus__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_9343_and__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),N) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% and_nat_def
tff(fact_9344_nat__times__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_times_as_int
tff(fact_9345_nat__minus__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_minus_as_int
tff(fact_9346_nat__div__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_div_as_int
tff(fact_9347_nat__mod__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X3),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_mod_as_int
tff(fact_9348_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,M)),N) ) ).

% less_eq_nat.simps(2)
tff(fact_9349_ln__neg__is__const,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_real(ln_ln_real,Xb) = the_real(aTP_Lamm_jm) ) ) ).

% ln_neg_is_const
tff(fact_9350_max__Suc2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_apv,N)),M) ).

% max_Suc2
tff(fact_9351_max__Suc1,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,N),aa_nat_fun_nat_nat(aTP_Lamm_apw,N)),M) ).

% max_Suc1
tff(fact_9352_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_9353_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( aa_int_nat(nat2,W) = M )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_9354_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = aa_int_nat(nat2,W) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_9355_split__nat,axiom,
    ! [P: fun_nat_o,I: int] :
      ( aa_nat_o(P,aa_int_nat(nat2,I))
    <=> ( ! [N8: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N8) )
           => aa_nat_o(P,N8) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_nat
tff(fact_9356_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_9357_nat__add__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_9358_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,N)),K) ) ) ).

% le_nat_iff
tff(fact_9359_Suc__as__int,axiom,
    ! [X3: nat] : aa_nat_nat(suc,X3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),one_one_int)) ).

% Suc_as_int
tff(fact_9360_nat__mult__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ).

% nat_mult_distrib
tff(fact_9361_nat__diff__distrib,axiom,
    ! [Z5: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z5),Z)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_9362_nat__diff__distrib_H,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_9363_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_9364_nat__div__distrib_H,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_9365_nat__div__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_9366_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),N)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),N) ) ) ).

% nat_power_eq
tff(fact_9367_nat__mod__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_9368_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% div_abs_eq_div_nat
tff(fact_9369_nat__floor__neg,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_9370_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% mod_abs_eq_div_nat
tff(fact_9371_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_9372_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(N),K)) ) ) ).

% take_bit_nat_eq
tff(fact_9373_floor__eq3,axiom,
    ! [N: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,N)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = N ) ) ) ).

% floor_eq3
tff(fact_9374_le__nat__floor,axiom,
    ! [Xb: nat,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A2))) ) ).

% le_nat_floor
tff(fact_9375_nat__2,axiom,
    aa_int_nat(nat2,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% nat_2
tff(fact_9376_diff__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_eb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)) ).

% diff_Suc
tff(fact_9377_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_9378_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),M)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,M)) ) ) ).

% nat_less_iff
tff(fact_9379_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_9380_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) ).

% nat_abs_int_diff
tff(fact_9381_floor__eq4,axiom,
    ! [N: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,N)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = N ) ) ) ).

% floor_eq4
tff(fact_9382_diff__nat__eq__if,axiom,
    ! [Z: int,Z5: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),zero_zero_int),
        aa_int_nat(nat2,Z),
        $let(
          d: int,
          d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z5),
          $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,d),zero_zero_int),zero_zero_nat,aa_int_nat(nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_9383_arccos__def,axiom,
    ! [Y: real] : aa_real_real(arccos,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_apx,Y)) ).

% arccos_def
tff(fact_9384_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),M)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M)),M = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_9385_even__nat__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ) ).

% even_nat_iff
tff(fact_9386_pi__half,axiom,
    aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))) = the_real(aTP_Lamm_apy) ).

% pi_half
tff(fact_9387_pi__def,axiom,
    pi = aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),the_real(aTP_Lamm_apy)) ).

% pi_def
tff(fact_9388_powr__real__of__int,axiom,
    ! [Xb: real,N: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),aa_int_real(ring_1_of_int_real,N)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),N),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,N)),aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,N))))) ) ) ).

% powr_real_of_int
tff(fact_9389_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_apz,Y)) ).

% arcsin_def
tff(fact_9390_floor__real__def,axiom,
    ! [Xb: real] : aa_real_int(archim6058952711729229775r_real,Xb) = the_int(aa_real_fun_int_o(aTP_Lamm_aqa,Xb)) ).

% floor_real_def
tff(fact_9391_htt__vebt__inserti__invar__vebt,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_nat_o(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Tb),Tib),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(vEBT_vebt_inserti,Tib),Xb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_insert(Tb,Xb))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))))))) ) ).

% htt_vebt_inserti_invar_vebt
tff(fact_9392_modulo__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = $ite(
        L = zero_zero_int,
        K,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,L)),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))))) ) ).

% modulo_int_def
tff(fact_9393_htt__vebt__buildupi_H,axiom,
    ! [N: nat] : aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(N))),aa_nat_nat(vEBT_V441764108873111860ildupi,N)) ).

% htt_vebt_buildupi'
tff(fact_9394_htt__vebt__buildupi,axiom,
    ! [N: nat] : aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(N))),aa_nat_nat(vEBT_V441764108873111860ildupi,N)) ).

% htt_vebt_buildupi
tff(fact_9395_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
     => aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(N))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U)) ) ).

% htt_vebt_buildupi'_univ
tff(fact_9396_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) )
     => aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(N))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U)) ) ).

% htt_vebt_buildupi_univ
tff(fact_9397_htt__vebt__inserti,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat_o(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Tb),Tib),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(vEBT_vebt_inserti,Tib),Xb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_insert(Tb,Xb))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb)))) ).

% htt_vebt_inserti
tff(fact_9398_sgn__mult__dvd__iff,axiom,
    ! [R: int,L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R)),L)),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_9399_mult__sgn__dvd__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R))),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_9400_dvd__sgn__mult__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R)),K))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_9401_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,R)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_9402_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V: option4927543243414619207at_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Vd: $o,Ve: $o] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(V,Va,Vb),Vc)),vEBT_Leafi((Vd),(Ve))) = bot_bot_assn ).

% vebt_assn_raw.simps(3)
tff(fact_9403_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_eq_sgn_abs
tff(fact_9404_GreatestI__nat,axiom,
    ! [P: fun_nat_o,K: nat,B2: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y2: nat] :
            ( aa_nat_o(P,Y2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),B2) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_nat
tff(fact_9405_Greatest__le__nat,axiom,
    ! [P: fun_nat_o,K: nat,B2: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y2: nat] :
            ( aa_nat_o(P,Y2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),B2) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),order_Greatest_nat(P)) ) ) ).

% Greatest_le_nat
tff(fact_9406_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_o,B2: nat] :
      ( ? [X_13: nat] : aa_nat_o(P,X_13)
     => ( ! [Y2: nat] :
            ( aa_nat_o(P,Y2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),B2) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_ex_nat
tff(fact_9407_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
       => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_9408_zsgn__def,axiom,
    ! [I: int] :
      aa_int_int(sgn_sgn_int,I) = $ite(
        I = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_9409_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_9410_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_9411_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K: int,Q2: int] :
      ( ( aa_int_int(sgn_sgn_int,R) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R)),aa_int_int(abs_abs_int,L))
       => ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L)),R) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_9412_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_9413_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K3) ) )
        | ? [L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4) ) )
        | ? [R6: int,L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R6) )
            & ( aa_int_int(sgn_sgn_int,R6) = aa_int_int(sgn_sgn_int,L4) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R6)),aa_int_int(abs_abs_int,L4))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4)),R6) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_9414_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R3) )
               => ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_9415_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,N: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( N = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N),M)))))) ) ).

% divide_int_unfold
tff(fact_9416_modulo__int__unfold,axiom,
    ! [K: int,M: nat,L: int,N: nat] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,N))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( N = zero_zero_nat ) ),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)),
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,N),M))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),N))))) ) ).

% modulo_int_unfold
tff(fact_9417_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = $ite(
        L = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))))) ) ).

% divide_int_def
tff(fact_9418_vebt__buildupi__rule,axiom,
    ! [N: nat] : aa_nat_o(time_htt_VEBT_VEBTi(pure_assn(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)),aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(N))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% vebt_buildupi_rule
tff(fact_9419_floor__rat__def,axiom,
    ! [Xb: rat] : archim3151403230148437115or_rat(Xb) = the_int(aTP_Lamm_aqb(Xb)) ).

% floor_rat_def
tff(fact_9420_sgn__div__eq__sgn__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) != zero_zero_int )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) = aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) ) ) ).

% sgn_div_eq_sgn_mult
tff(fact_9421_highi__hT,axiom,
    ! [Xb: nat,N: nat] : aa_nat_o(time_htt_nat(one_one_assn,aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Xb),N),aa_nat_fun_nat_assn(aTP_Lamm_aqc(Xb),N)),one_one_nat) ).

% highi_hT
tff(fact_9422_lowi__hT,axiom,
    ! [Xb: nat,N: nat] : aa_nat_o(time_htt_nat(one_one_assn,aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Xb),N),aa_nat_fun_nat_assn(aTP_Lamm_aqd(Xb),N)),one_one_nat) ).

% lowi_hT
tff(fact_9423_vebt__minti__hT,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi] : aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Tb),Tib),aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Tib),aa_VEB7611521244314475576t_assn(aTP_Lamm_aqe(Tb),Tib)),one_one_nat) ).

% vebt_minti_hT
tff(fact_9424_vebt__maxti__hT,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi] : aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Tb),Tib),aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Tib),aa_VEB7611521244314475576t_assn(aTP_Lamm_aqf(Tb),Tib)),one_one_nat) ).

% vebt_maxti_hT
tff(fact_9425_minNrulli__ruleT,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi] : aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Tb),Tib),aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Tib),aa_VEB6687300313434771872o_assn(aTP_Lamm_aqg(Tb),Tib)),one_one_nat) ).

% minNrulli_ruleT
tff(fact_9426_zero__le__sgn__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% zero_le_sgn_iff
tff(fact_9427_sgn__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sgn_sgn_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% sgn_le_0_iff
tff(fact_9428_htt__vebt__memberi,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Tb),Tib),aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(vEBT_vebt_memberi,Tib),Xb),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_aqh(Tb),Tib),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb)))) ).

% htt_vebt_memberi
tff(fact_9429_htt__vebt__memberi__invar__vebt,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Tb),Tib),aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(vEBT_vebt_memberi,Tib),Xb),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_aqh(Tb),Tib),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))))))) ) ).

% htt_vebt_memberi_invar_vebt
tff(fact_9430_htt__vebt__succi,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Tb),Tib),aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_succi,Tib),Xb),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_aqi(Tb),Tib),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))))))) ) ).

% htt_vebt_succi
tff(fact_9431_sgn__rat__def,axiom,
    ! [A2: rat] :
      sgn_sgn_rat(A2) = $ite(
        A2 = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2),one_one_rat,uminus_uminus_rat(one_one_rat)) ) ).

% sgn_rat_def
tff(fact_9432_obtain__pos__sum,axiom,
    ! [R: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R)
     => ~ ! [S2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S2)
           => ! [T: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T)
               => ( R != plus_plus_rat(S2,T) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9433_less__eq__rat__def,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_rat_def
tff(fact_9434_abs__rat__def,axiom,
    ! [A2: rat] :
      abs_abs_rat(A2) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat),uminus_uminus_rat(A2),A2) ).

% abs_rat_def
tff(fact_9435_sgn__root,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)) = aa_real_real(sgn_sgn_real,Xb) ) ) ).

% sgn_root
tff(fact_9436_cis__Arg,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_complex(cis,aa_complex_real(arg,Z)) = aa_complex_complex(sgn_sgn_complex,Z) ) ) ).

% cis_Arg
tff(fact_9437_sgn__real__def,axiom,
    ! [A2: real] :
      aa_real_real(sgn_sgn_real,A2) = $ite(
        A2 = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_real_def
tff(fact_9438_sgn__integer__code,axiom,
    ! [K: code_integer] :
      aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = $ite(
        K = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) ) ).

% sgn_integer_code
tff(fact_9439_sgn__power__injE,axiom,
    ! [A2: real,N: nat,Xb: real,B2: real] :
      ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),N)) = Xb )
     => ( ( Xb = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,B2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B2)),N)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_9440_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_real(aa_nat_fun_real_real(root,N),aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Y)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y)),N))) = Y ) ) ).

% root_sgn_power
tff(fact_9441_sgn__power__root,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb))),N)) = Xb ) ) ).

% sgn_power_root
tff(fact_9442_cis__Arg__unique,axiom,
    ! [Z: complex,Xb: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,Xb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => ( aa_complex_real(arg,Z) = Xb ) ) ) ) ).

% cis_Arg_unique
tff(fact_9443_split__root,axiom,
    ! [P: fun_real_o,N: nat,Xb: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,N),Xb))
    <=> ( ( ( N = zero_zero_nat )
         => aa_real_o(P,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
         => ! [Y3: real] :
              ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Y3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y3)),N)) = Xb )
             => aa_real_o(P,Y3) ) ) ) ) ).

% split_root
tff(fact_9444_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,aa_complex_real(arg,Z)) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ) ).

% Arg_correct
tff(fact_9445_arctan__inverse,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),Xb)) = aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Xb)),pi)),numeral_numeral_real(aa_num_num(bit0,one)))),aa_real_real(arctan,Xb)) ) ) ).

% arctan_inverse
tff(fact_9446_vebt__pred_H__rf__abstr,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Tb),Tib),aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_predi,Tb),Tib),Xb),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_aqj(Tb),Tib),Xb)) ) ).

% vebt_pred'_rf_abstr
tff(fact_9447_vebt__succi_H__rf__abstr,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),N)
     => hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Tb),Tib),aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_succi,Tb),Tib),Xb),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_aqi(Tb),Tib),Xb)) ) ).

% vebt_succi'_rf_abstr
tff(fact_9448_vebt__memberi_H__rf__abstr,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : hoare_hoare_triple_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Tb),Tib),aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(vEBT_V854960066525838166emberi,Tb),Tib),Xb),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_aqh(Tb),Tib),Xb)) ).

% vebt_memberi'_rf_abstr
tff(fact_9449_vebt__minti__h,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Tb),Tib),aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Tib),aa_VEB7611521244314475576t_assn(aTP_Lamm_aqe(Tb),Tib)) ).

% vebt_minti_h
tff(fact_9450_vebt__maxti__h,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Tb),Tib),aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Tib),aa_VEB7611521244314475576t_assn(aTP_Lamm_aqf(Tb),Tib)) ).

% vebt_maxti_h
tff(fact_9451_minNulli__rule,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi] : hoare_hoare_triple_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Tb),Tib),aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Tib),aa_VEB6687300313434771872o_assn(aTP_Lamm_aqg(Tb),Tib)) ).

% minNulli_rule
tff(fact_9452_vebt__maxtilist,axiom,
    ! [I: nat,Ts: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ts))
     => hoare_7629718768684598413on_nat(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Ts,Tsi),aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,nth_VEBT_VEBTi(Tsi,I)),aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_aqk(I),Ts),Tsi)) ) ).

% vebt_maxtilist
tff(fact_9453_vebt__mintilist,axiom,
    ! [I: nat,Ts: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ts))
     => hoare_7629718768684598413on_nat(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Ts,Tsi),aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,nth_VEBT_VEBTi(Tsi,I)),aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_aql(I),Ts),Tsi)) ) ).

% vebt_mintilist
tff(fact_9454_vebt__inserti_H__rf__abstr,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Tb),Tib),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(vEBT_V3964819847710782039nserti,Tb),Tib),Xb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_insert(Tb,Xb))) ).

% vebt_inserti'_rf_abstr
tff(fact_9455_highi__h,axiom,
    ! [Xb: nat,N: nat] : hoare_3067605981109127869le_nat(one_one_assn,aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Xb),N),aa_nat_fun_nat_assn(aTP_Lamm_aqc(Xb),N)) ).

% highi_h
tff(fact_9456_lowi__h,axiom,
    ! [Xb: nat,N: nat] : hoare_3067605981109127869le_nat(one_one_assn,aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Xb),N),aa_nat_fun_nat_assn(aTP_Lamm_aqd(Xb),N)) ).

% lowi_h
tff(fact_9457_builupi_Hcorr,axiom,
    ! [N: nat] : hoare_1429296392585015714_VEBTi(one_one_assn,aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(N))) ).

% builupi'corr
tff(fact_9458_builupicorr,axiom,
    ! [N: nat] : hoare_1429296392585015714_VEBTi(one_one_assn,aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,N),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_buildup(N))) ).

% builupicorr
tff(fact_9459_Arg__def,axiom,
    ! [Z: complex] :
      aa_complex_real(arg,Z) = $ite(Z = zero_zero_complex,zero_zero_real,fChoice_real(aTP_Lamm_aqm(Z))) ).

% Arg_def
tff(fact_9460_assert_H__rule,axiom,
    ! [P: assn,Phi: $o] :
      ( ! [H5: produc3658429121746597890et_nat] :
          ( rep_assn(P,H5)
         => (Phi) )
     => hoare_8945653483474564448t_unit(P,refine_Imp_assert((Phi)),aTP_Lamm_aqn(P)) ) ).

% assert'_rule
tff(fact_9461_vebt__assn__raw_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: vEBT_VEBTi,Y: assn] :
      ( ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xb),Xaa) = Y )
     => ( ! [A: $o,B: $o] :
            ( ( Xb = vEBT_Leaf((A),(B)) )
           => ! [Ai: $o,Bi: $o] :
                ( ( Xaa = vEBT_Leafi((Ai),(Bi)) )
               => ( Y != pure_assn(( ( (Ai) = (A) )
                      & ( (Bi) = (B) ) )) ) ) )
       => ( ! [Mmo: option4927543243414619207at_nat,Deg: nat,Tree_list: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg,Tree_list),Summary) )
             => ! [Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
                  ( ( Xaa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi) )
                 => ( Y != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                          aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                              pure_assn(( ( Mmoi = Mmo )
                                & ( Degi = Deg ) ))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Summaryi))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_aqo(Tree_list),Tree_array))) ) ) )
         => ( ( ? [V2: option4927543243414619207at_nat,Va2: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] : Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va2,Vb3),Vc3)
             => ( ? [Vd3: $o,Ve3: $o] : Xaa = vEBT_Leafi((Vd3),(Ve3))
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ? [Vd3: $o,Ve3: $o] : Xb = vEBT_Leaf((Vd3),(Ve3))
               => ( ? [V2: option4927543243414619207at_nat,Va2: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] : Xaa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va2,Vb3),Vc3)
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% vebt_assn_raw.elims
tff(fact_9462_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [Mmo2: option4927543243414619207at_nat,Deg2: nat,Tree_list2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
      aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo2,Deg2,Tree_list2),Summary2)),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi2,Degi2,Tree_array2),Summaryi2)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
          aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
              pure_assn(( ( Mmoi2 = Mmo2 )
                & ( Degi2 = Deg2 ) ))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),Summaryi2))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_aqo(Tree_list2),Tree_array2))) ).

% vebt_assn_raw.simps(2)
tff(fact_9463_vebt__assn__raw_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: vEBT_VEBTi,Y: assn] :
      ( ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xb),Xaa) = Y )
     => ( accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(Xb,Xaa))
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ! [Ai: $o,Bi: $o] :
                  ( ( Xaa = vEBT_Leafi((Ai),(Bi)) )
                 => ( ( Y = pure_assn(( ( (Ai) = (A) )
                          & ( (Bi) = (B) ) )) )
                   => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(vEBT_Leaf((A),(B)),vEBT_Leafi((Ai),(Bi)))) ) ) )
         => ( ! [Mmo: option4927543243414619207at_nat,Deg: nat,Tree_list: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg,Tree_list),Summary) )
               => ! [Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
                    ( ( Xaa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi) )
                   => ( ( Y = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                              aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,
                                  pure_assn(( ( Mmoi = Mmo )
                                    & ( Degi = Deg ) ))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Summaryi))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_aqo(Tree_list),Tree_array))) )
                     => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg,Tree_list),Summary),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi))) ) ) )
           => ( ! [V2: option4927543243414619207at_nat,Va2: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va2,Vb3),Vc3) )
                 => ! [Vd3: $o,Ve3: $o] :
                      ( ( Xaa = vEBT_Leafi((Vd3),(Ve3)) )
                     => ( ( Y = bot_bot_assn )
                       => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va2,Vb3),Vc3),vEBT_Leafi((Vd3),(Ve3)))) ) ) )
             => ~ ! [Vd3: $o,Ve3: $o] :
                    ( ( Xb = vEBT_Leaf((Vd3),(Ve3)) )
                   => ! [V2: option4927543243414619207at_nat,Va2: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                        ( ( Xaa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va2,Vb3),Vc3) )
                       => ( ( Y = bot_bot_assn )
                         => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(vEBT_Leaf((Vd3),(Ve3)),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va2,Vb3),Vc3))) ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.pelims
tff(fact_9464_assnle,axiom,
    ! [TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary2: vEBT_VEBT,X14: vEBT_VEBTi] : aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14)))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14)),snga_assn_VEBT_VEBTi(X13,Tree_is))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))) ).

% assnle
tff(fact_9465_big__assn__simp,axiom,
    ! [H: nat,TreeList: list_VEBT_VEBT,L: nat,Xb: vEBT_VEBTi,Xaa: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,H),size_s6755466524823107622T_VEBT(TreeList))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),Xb)),pure_assn(Xaa = vEBT_vebt_mint(vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L))))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H,Xb))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(H,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is)))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H,Xb))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),pure_assn(Xaa = vEBT_vebt_mint(vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L))))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,H,vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L)),list_u6098035379799741383_VEBTi(Tree_is,H,Xb)))) ) ).

% big_assn_simp
tff(fact_9466_big__assn__simp_H,axiom,
    ! [H: nat,TreeList: list_VEBT_VEBT,Xaa: vEBT_VEBT,L: nat,Xb: vEBT_VEBTi,Xba: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,H),size_s6755466524823107622T_VEBT(TreeList))
     => ( ( Xaa = vEBT_vebt_delete(nth_VEBT_VEBT(TreeList,H),L) )
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xaa),Xb)),pure_assn(Xba = vEBT_vebt_mint(Xaa)))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H,Xb))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(TreeList)),insert_nat(H,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeList,Tree_is)))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,list_u6098035379799741383_VEBTi(Tree_is,H,Xb))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),pure_assn(Xba = vEBT_vebt_mint(Xaa)))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeList,H,Xaa),list_u6098035379799741383_VEBTi(Tree_is,H,Xb)))) ) ) ).

% big_assn_simp'
tff(fact_9467_local_Oext,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),size_s6755466524823107622T_VEBT(TreeList))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,Y)),nth_VEBT_VEBTi(Tree_is,Y))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(aa_nat1818925681583257441et_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))))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(aa_nat1818925681583257441et_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))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,Y)),nth_VEBT_VEBTi(Tree_is,Y)))) ) ).

% local.ext
tff(fact_9468_recomp,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,I)),nth_VEBT_VEBTi(Tree_is,I))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(aa_nat1818925681583257441et_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))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14)),snga_assn_VEBT_VEBTi(X13,Tree_is))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))) ) ).

% recomp
tff(fact_9469_repack,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,Rest: assn,X13: array_VEBT_VEBTi,Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeList))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,I)),nth_VEBT_VEBTi(Tree_is,I))),Rest)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(X13,Tree_is)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(aa_nat1818925681583257441et_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)))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Rest),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),snga_assn_VEBT_VEBTi(X13,Tree_is))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))) ) ).

% repack
tff(fact_9470_txe,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),size_s6755466524823107622T_VEBT(TreeList))
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,nth_VEBT_VEBT(TreeList,Y)),nth_VEBT_VEBTi(Tree_is,Y))),snga_assn_VEBT_VEBTi(X13,Tree_is))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14))),vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(aa_nat1818925681583257441et_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))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary2),X14)),snga_assn_VEBT_VEBTi(X13,Tree_is))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeList,Tree_is))) ) ).

% txe
tff(fact_9471_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_o] :
      ( ? [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M6),N)
          & aa_nat_o(P,M6) )
    <=> ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N))
          & aa_nat_o(P,X) ) ) ).

% ex_nat_less_eq
tff(fact_9472_all__nat__less__eq,axiom,
    ! [N: nat,P: fun_nat_o] :
      ( ! [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M6),N)
         => aa_nat_o(P,M6) )
    <=> ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N))
         => aa_nat_o(P,X) ) ) ).

% all_nat_less_eq
tff(fact_9473_lessThan__atLeast0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N) ).

% lessThan_atLeast0
tff(fact_9474_atLeastLessThan0,axiom,
    ! [M: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_9475_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(N,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)) ).

% atLeast0_lessThan_Suc
tff(fact_9476_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N2: set_nat,N: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N))
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9477_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_nat_nat(suc,N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N),insert_nat(N,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_9478_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : groups708209901874060359at_nat(suc,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,zero_zero_nat)),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_Suc_fact
tff(fact_9479_prod__Suc__fact,axiom,
    ! [N: nat] : groups708209901874060359at_nat(suc,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)) = aa_nat_nat(semiri1408675320244567234ct_nat,N) ).

% prod_Suc_fact
tff(fact_9480_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_num_nat(numeral_numeral_nat,K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)),insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),aa_num_nat(pred_numeral,K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_9481_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(suc,zero_zero_nat)),N) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,N),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9482_sum__power2,axiom,
    ! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),one_one_nat) ).

% sum_power2
tff(fact_9483_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_eb),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,M),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9484_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A2: fun_nat_nat,B2: fun_nat_nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),N)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A2,I2)),aa_nat_nat(A2,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),N)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B2,J2)),aa_nat_nat(B2,I2)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aqp(A2),B2)),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A2),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N))),aa_set_nat_nat(groups3542108847815614940at_nat(B2),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9485_root__def,axiom,
    ! [N: nat,Xb: real] :
      aa_real_real(aa_nat_fun_real_real(root,N),Xb) = $ite(N = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_aqq,N)),Xb)) ).

% root_def
tff(fact_9486_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_9487_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa_set_int_o(finite_finite_int,set_or4662586982721622107an_int(zero_zero_int,U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_9488_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(L,aa_int_int(aa_int_fun_int_int(plus_plus_int,U),one_one_int)) = set_or1266510415728281911st_int(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_9489_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),K)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,N))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_9490_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% not_negative_int_iff
tff(fact_9491_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% not_nonnegative_int_iff
tff(fact_9492_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri631733984087533419it_int(N),K))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_9493_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(N),K)),zero_zero_int)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) ).

% signed_take_bit_negative_iff
tff(fact_9494_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_nat_nat(suc,N))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),N) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_9495_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_nat_nat(suc,N))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),N) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_9496_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N: num] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_num_nat(numeral_numeral_nat,N))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,N)) ) ).

% bit_minus_numeral_int(1)
tff(fact_9497_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N: num] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_num_nat(numeral_numeral_nat,N))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,N)) ) ).

% bit_minus_numeral_int(2)
tff(fact_9498_bin__nth__minus__Bit0,axiom,
    ! [N: nat,W: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))),N)
      <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% bin_nth_minus_Bit0
tff(fact_9499_bin__nth__minus__Bit1,axiom,
    ! [N: nat,W: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))),N)
      <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% bin_nth_minus_Bit1
tff(fact_9500_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,K)),N)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))),N) ) ).

% bit_minus_int_iff
tff(fact_9501_bit__not__int__iff,axiom,
    ! [K: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),N)
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) ).

% bit_not_int_iff
tff(fact_9502_bit__and__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),N)
    <=> ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),N) ) ) ).

% bit_and_int_iff
tff(fact_9503_not__int__def,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int) ).

% not_int_def
tff(fact_9504_and__not__numerals_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = zero_zero_int ).

% and_not_numerals(1)
tff(fact_9505_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int)),N)
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) ).

% bit_not_int_iff'
tff(fact_9506_not__int__div__2,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% not_int_div_2
tff(fact_9507_even__not__iff__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,K))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).

% even_not_iff_int
tff(fact_9508_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(4)
tff(fact_9509_and__not__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_9510_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(M),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_9511_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N: num,Q2: num] :
      ( ( aa_num_option_num(bit_and_not_num(M),N) = aa_num_option_num(some_num,Q2) )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = aa_num_int(numeral_numeral_int,Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_9512_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N3: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N3),M2)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),M2)
              <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N3) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N3),one_one_nat))
              <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N3) ) ) ) ).

% int_bit_bound
tff(fact_9513_and__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% and_not_numerals(5)
tff(fact_9514_and__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(7)
tff(fact_9515_and__not__numerals_I3_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_9516_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( aa_num_option_num(bit_and_not_num(M),N) = none_num )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_9517_and__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% and_not_numerals(9)
tff(fact_9518_and__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% and_not_numerals(6)
tff(fact_9519_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(N),M)) ).

% int_numeral_not_and_num
tff(fact_9520_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = aa_option_num_int(case_option_int_num(zero_zero_int,numeral_numeral_int),aa_num_option_num(bit_and_not_num(M),N)) ).

% int_numeral_and_not_num
tff(fact_9521_bit__int__def,axiom,
    ! [K: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ) ).

% bit_int_def
tff(fact_9522_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% and_not_numerals(8)
tff(fact_9523_not__int__rec,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_9524_Bit__Operations_Oset__bit__eq,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se7879613467334960850it_int(N),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% Bit_Operations.set_bit_eq
tff(fact_9525_unset__bit__eq,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se4203085406695923979it_int(N),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))) ).

% unset_bit_eq
tff(fact_9526_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N)))),aa_int_int(bit_se2923211474154528505it_int(N),K)) ).

% take_bit_Suc_from_most
tff(fact_9527_int__not__code_I1_J,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% int_not_code(1)
tff(fact_9528_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = $ite(
        K = aa_int_int(uminus_uminus_int,one_one_int),
        aa_int_int(bit_ri7919022796975470100ot_int,L),
        $ite(
          L = aa_int_int(uminus_uminus_int,one_one_int),
          aa_int_int(bit_ri7919022796975470100ot_int,K),
          $ite(
            K = zero_zero_int,
            L,
            $ite(L = zero_zero_int,K,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ) ).

% xor_int_unfold
tff(fact_9529_num_Osize__gen_I3_J,axiom,
    ! [X33: num] : aa_num_nat(size_num,aa_num_num(bit1,X33)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X33)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_9530_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_9531_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% or_negative_int_iff
tff(fact_9532_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_9533_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)),zero_zero_int)
    <=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% xor_negative_int_iff
tff(fact_9534_or__minus__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).

% or_minus_numerals(2)
tff(fact_9535_or__minus__numerals_I6_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N))) ).

% or_minus_numerals(6)
tff(fact_9536_or__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(2)
tff(fact_9537_or__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_nat_numerals(4)
tff(fact_9538_or__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_9539_or__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_nat_numerals(3)
tff(fact_9540_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,N)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_9541_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,N)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_9542_xor__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),aa_int_int(bit_ri7919022796975470100ot_int,L))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),L)) ).

% xor_int_def
tff(fact_9543_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)) ) ).

% or_greater_eq
tff(fact_9544_OR__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Y)) ) ) ).

% OR_lower
tff(fact_9545_bit__or__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)),N)
    <=> ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N)
        | aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),N) ) ) ).

% bit_or_int_iff
tff(fact_9546_bit__xor__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)),N)
    <=> ~ ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N)
        <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),N) ) ) ).

% bit_xor_int_iff
tff(fact_9547_or__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% or_nat_def
tff(fact_9548_int__or__code_I1_J,axiom,
    ! [J: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,zero_zero_int),J) = J ).

% int_or_code(1)
tff(fact_9549_int__or__code_I2_J,axiom,
    ! [I: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,I),zero_zero_int) = I ).

% int_or_code(2)
tff(fact_9550_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,N)) ).

% not_bit_Suc_0_Suc
tff(fact_9551_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),N)
    <=> ( N = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_9552_XOR__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xb),Y)) ) ) ).

% XOR_lower
tff(fact_9553_int__xor__code_I2_J,axiom,
    ! [I: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,I),zero_zero_int) = I ).

% int_xor_code(2)
tff(fact_9554_int__xor__code_I1_J,axiom,
    ! [J: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,zero_zero_int),J) = J ).

% int_xor_code(1)
tff(fact_9555_plus__and__or,axiom,
    ! [Xb: int,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) ).

% plus_and_or
tff(fact_9556_or__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_int_int(bit_ri7919022796975470100ot_int,L))) ).

% or_int_def
tff(fact_9557_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,N)) ).

% not_bit_Suc_0_numeral
tff(fact_9558_or__not__numerals_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(1)
tff(fact_9559_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_9560_or__not__numerals_I2_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).

% or_not_numerals(2)
tff(fact_9561_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),N)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) ) ).

% bit_nat_iff
tff(fact_9562_or__not__numerals_I3_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N))) ).

% or_not_numerals(3)
tff(fact_9563_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9564_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_9565_bit__nat__def,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,M),N)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ) ).

% bit_nat_def
tff(fact_9566_or__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% or_not_numerals(6)
tff(fact_9567_XOR__upper,axiom,
    ! [Xb: int,N: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xb),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ) ) ).

% XOR_upper
tff(fact_9568_OR__upper,axiom,
    ! [Xb: int,N: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ) ) ) ).

% OR_upper
tff(fact_9569_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(5)
tff(fact_9570_test__bit__int__code_I1_J,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,zero_zero_int),N) ).

% test_bit_int_code(1)
tff(fact_9571_int__and__code_I1_J,axiom,
    ! [J: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,zero_zero_int),J) = zero_zero_int ).

% int_and_code(1)
tff(fact_9572_int__and__code_I2_J,axiom,
    ! [I: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,I),zero_zero_int) = zero_zero_int ).

% int_and_code(2)
tff(fact_9573_or__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% or_Suc_0_eq
tff(fact_9574_Suc__0__or__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% Suc_0_or_eq
tff(fact_9575_or__nat__rec,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_9576_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(8)
tff(fact_9577_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))))) ).

% or_not_numerals(9)
tff(fact_9578_xor__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) != ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_9579_or__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            | ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_9580_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),N) = $ite(
        M = zero_zero_nat,
        N,
        $ite(N = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_9581_or__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = $ite(
        ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
        | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) ),
        aa_int_int(uminus_uminus_int,one_one_int),
        $ite(
          K = zero_zero_int,
          L,
          $ite(L = zero_zero_int,K,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% or_int_unfold
tff(fact_9582_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : aa_num_nat(size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_9583_or__minus__numerals_I5_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bitM,N)))) ).

% or_minus_numerals(5)
tff(fact_9584_or__minus__numerals_I1_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bitM,N)))) ).

% or_minus_numerals(1)
tff(fact_9585_xor__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb)) ).

% xor_nat_numerals(4)
tff(fact_9586_xor__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% xor_nat_numerals(3)
tff(fact_9587_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y)) ).

% xor_nat_numerals(2)
tff(fact_9588_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_9589_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),aa_num_num(bit0,N)))) ).

% or_minus_numerals(8)
tff(fact_9590_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),aa_num_num(bit0,N)))) ).

% or_minus_numerals(4)
tff(fact_9591_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),aa_num_num(bitM,N)))) ).

% or_minus_numerals(7)
tff(fact_9592_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,N)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),aa_num_num(bitM,N)))) ).

% or_minus_numerals(3)
tff(fact_9593_or__not__num__neg_Osimps_I1_J,axiom,
    aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),one) = one ).

% or_not_num_neg.simps(1)
tff(fact_9594_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,N)),one) = aa_num_num(bit0,one) ).

% or_not_num_neg.simps(4)
tff(fact_9595_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,N)),aa_num_num(bit1,M)) = aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).

% or_not_num_neg.simps(6)
tff(fact_9596_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,N)),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_9597_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit1,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(3)
tff(fact_9598_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit0,N)),aa_num_num(bit0,M)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).

% or_not_num_neg.simps(5)
tff(fact_9599_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,N)),aa_num_num(bit1,M)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).

% or_not_num_neg.simps(9)
tff(fact_9600_xor__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,N))) ).

% xor_nat_def
tff(fact_9601_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,one),aa_num_num(bit0,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(2)
tff(fact_9602_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N: num,M: num] : aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,aa_num_num(bit1,N)),aa_num_num(bit0,M)) = aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M)) ).

% or_not_num_neg.simps(8)
tff(fact_9603_or__not__num__neg_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y: num] :
      ( ( aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,Xb),Xaa) = Y )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Y != one ) ) )
       => ( ( ( Xb = one )
           => ! [M3: num] :
                ( ( Xaa = aa_num_num(bit0,M3) )
               => ( Y != aa_num_num(bit1,M3) ) ) )
         => ( ( ( Xb = one )
             => ! [M3: num] :
                  ( ( Xaa = aa_num_num(bit1,M3) )
                 => ( Y != aa_num_num(bit1,M3) ) ) )
           => ( ( ? [N3: num] : Xb = aa_num_num(bit0,N3)
               => ( ( Xaa = one )
                 => ( Y != aa_num_num(bit0,one) ) ) )
             => ( ! [N3: num] :
                    ( ( Xb = aa_num_num(bit0,N3) )
                   => ! [M3: num] :
                        ( ( Xaa = aa_num_num(bit0,M3) )
                       => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N3),M3)) ) ) )
               => ( ! [N3: num] :
                      ( ( Xb = aa_num_num(bit0,N3) )
                     => ! [M3: num] :
                          ( ( Xaa = aa_num_num(bit1,M3) )
                         => ( Y != aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N3),M3)) ) ) )
                 => ( ( ? [N3: num] : Xb = aa_num_num(bit1,N3)
                     => ( ( Xaa = one )
                       => ( Y != one ) ) )
                   => ( ! [N3: num] :
                          ( ( Xb = aa_num_num(bit1,N3) )
                         => ! [M3: num] :
                              ( ( Xaa = aa_num_num(bit0,M3) )
                             => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N3),M3)) ) ) )
                     => ~ ! [N3: num] :
                            ( ( Xb = aa_num_num(bit1,N3) )
                           => ! [M3: num] :
                                ( ( Xaa = aa_num_num(bit1,M3) )
                               => ( Y != aa_num_num(bitM,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N3),M3)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_9604_numeral__or__not__num__eq,axiom,
    ! [M: num,N: num] : aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),N)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N)))) ).

% numeral_or_not_num_eq
tff(fact_9605_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,N),M))) ).

% int_numeral_not_or_num_neg
tff(fact_9606_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(bit_or_not_num_neg,M),N))) ).

% int_numeral_or_not_num_neg
tff(fact_9607_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = $ite(
        M = zero_zero_nat,
        N,
        $ite(N = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_9608_xor__nat__rec,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_9609_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% Suc_0_xor_eq
tff(fact_9610_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) ).

% xor_Suc_0_eq
tff(fact_9611_int__lsb__numeral_I1_J,axiom,
    ~ aa_int_o(least_4859182151741483524sb_int,zero_zero_int) ).

% int_lsb_numeral(1)
tff(fact_9612_int__lsb__numeral_I2_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,one_one_int) ).

% int_lsb_numeral(2)
tff(fact_9613_int__lsb__numeral_I6_J,axiom,
    ! [W: num] : ~ aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) ).

% int_lsb_numeral(6)
tff(fact_9614_int__lsb__numeral_I3_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,one)) ).

% int_lsb_numeral(3)
tff(fact_9615_int__lsb__numeral_I7_J,axiom,
    ! [W: num] : aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))) ).

% int_lsb_numeral(7)
tff(fact_9616_int__lsb__numeral_I4_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% int_lsb_numeral(4)
tff(fact_9617_int__lsb__numeral_I8_J,axiom,
    ! [W: num] : ~ aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))) ).

% int_lsb_numeral(8)
tff(fact_9618_int__lsb__numeral_I5_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) ).

% int_lsb_numeral(5)
tff(fact_9619_int__lsb__numeral_I9_J,axiom,
    ! [W: num] : aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))) ).

% int_lsb_numeral(9)
tff(fact_9620_lsb__int__def,axiom,
    ! [I: int] :
      ( aa_int_o(least_4859182151741483524sb_int,I)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,I),zero_zero_nat) ) ).

% lsb_int_def
tff(fact_9621_bin__last__conv__lsb,axiom,
    aTP_Lamm_aqr = least_4859182151741483524sb_int ).

% bin_last_conv_lsb
tff(fact_9622_pow_Osimps_I1_J,axiom,
    ! [Xb: num] : aa_num_num(aa_num_fun_num_num(pow,Xb),one) = Xb ).

% pow.simps(1)
tff(fact_9623_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( aa_set_real_o(member_real(N),ring_1_Ints_real)
     => ( aa_real_complex(cis,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)),N)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_9624_sin__times__pi__eq__0,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,aa_real_real(times_times_real(Xb),pi)) = zero_zero_real )
    <=> aa_set_real_o(member_real(Xb),ring_1_Ints_real) ) ).

% sin_times_pi_eq_0
tff(fact_9625_sin__integer__2pi,axiom,
    ! [N: real] :
      ( aa_set_real_o(member_real(N),ring_1_Ints_real)
     => ( aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)),N)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_9626_cos__integer__2pi,axiom,
    ! [N: real] :
      ( aa_set_real_o(member_real(N),ring_1_Ints_real)
     => ( aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)),N)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_9627_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(inverse_inverse_rat(P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_aqs),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_9628_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(numeral_numeral_rat(K)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% quotient_of_number(3)
tff(fact_9629_rat__one__code,axiom,
    quotient_of(one_one_rat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int) ).

% rat_one_code
tff(fact_9630_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% rat_zero_code
tff(fact_9631_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(uminus_uminus_rat(numeral_numeral_rat(K))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int) ).

% quotient_of_number(5)
tff(fact_9632_quotient__of__number_I4_J,axiom,
    quotient_of(uminus_uminus_rat(one_one_rat)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% quotient_of_number(4)
tff(fact_9633_quotient__of__denom__pos,axiom,
    ! [R: rat,P2: int,Q2: int] :
      ( ( quotient_of(R) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q2) ) ).

% quotient_of_denom_pos
tff(fact_9634_quotient__of__denom__pos_H,axiom,
    ! [R: rat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,quotient_of(R))) ).

% quotient_of_denom_pos'
tff(fact_9635_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(uminus_uminus_rat(P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_aqt),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_9636_rat__less__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),Q2)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_aqv(Q2)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_9637_rat__floor__code,axiom,
    ! [P2: rat] : archim3151403230148437115or_rat(P2) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(divide_divide_int),quotient_of(P2)) ).

% rat_floor_code
tff(fact_9638_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(abs_abs_rat(P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_aqw),quotient_of(P2)) ).

% rat_abs_code
tff(fact_9639_rat__less__eq__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),Q2)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_aqy(Q2)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_9640_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(sgn_sgn_rat(P2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(sgn_sgn_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,quotient_of(P2)))),one_one_int) ).

% rat_sgn_code
tff(fact_9641_quotient__of__int,axiom,
    ! [A2: int] : quotient_of(aa_int_rat(of_int,A2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),one_one_int) ).

% quotient_of_int
tff(fact_9642_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(plus_plus_rat(P2,Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ara(Q2)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_9643_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(minus_minus_rat(P2,Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_arc(Q2)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_9644_normalize__denom__zero,axiom,
    ! [P2: int] : aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),zero_zero_int)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% normalize_denom_zero
tff(fact_9645_normalize__negative,axiom,
    ! [Q2: int,P2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Q2),zero_zero_int)
     => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,P2)),aa_int_int(uminus_uminus_int,Q2))) ) ) ).

% normalize_negative
tff(fact_9646_normalize__denom__pos,axiom,
    ! [R: product_prod_int_int,P2: int,Q2: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,R) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q2) ) ).

% normalize_denom_pos
tff(fact_9647_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P2: int,R: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R),S)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P2),S) = aa_int_int(aa_int_fun_int_int(times_times_int,R),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9648_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(divide_divide_rat(P2,Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_are(Q2)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_9649_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(times_times_rat(P2),Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_arg(Q2)),quotient_of(P2)) ).

% rat_times_code
tff(fact_9650_normalize__def,axiom,
    ! [P2: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(normalize,P2) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),
        $let(
          a: int,
          a:= aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),a)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),a)) ),
        $ite(
          aa_Pro6799258434465214121nt_int(product_snd_int_int,P2) = zero_zero_int,
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),
          $let(
            a: int,
            a:= aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2))),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),a)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),a)) ) ) ) ).

% normalize_def
tff(fact_9651_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),aa_num_int(numeral_numeral_int,K))) = divide_divide_rat(one_one_rat,numeral_numeral_rat(K)) ).

% Frct_code_post(5)
tff(fact_9652_gcd__1__int,axiom,
    ! [M: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),one_one_int) = one_one_int ).

% gcd_1_int
tff(fact_9653_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N))
    <=> ( ( M != zero_zero_int )
        | ( N != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_9654_gcd__neg__numeral__2__int,axiom,
    ! [Xb: int,N: num] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),aa_num_int(numeral_numeral_int,N)) ).

% gcd_neg_numeral_2_int
tff(fact_9655_gcd__neg__numeral__1__int,axiom,
    ! [N: num,Xb: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),Xb) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_num_int(numeral_numeral_int,N)),Xb) ).

% gcd_neg_numeral_1_int
tff(fact_9656_gcd__0__left__int,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,zero_zero_int),Xb) = aa_int_int(abs_abs_int,Xb) ).

% gcd_0_left_int
tff(fact_9657_gcd__0__int,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),zero_zero_int) = aa_int_int(abs_abs_int,Xb) ).

% gcd_0_int
tff(fact_9658_gcd__ge__0__int,axiom,
    ! [Xb: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Y)) ).

% gcd_ge_0_int
tff(fact_9659_gcd__le1__int,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A2),B2)),A2) ) ).

% gcd_le1_int
tff(fact_9660_gcd__le2__int,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A2),B2)),B2) ) ).

% gcd_le2_int
tff(fact_9661_gcd__cases__int,axiom,
    ! [Xb: int,Y: int,P: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Y)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),aa_int_int(uminus_uminus_int,Y))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,Xb)),Y)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,Xb)),aa_int_int(uminus_uminus_int,Y))) ) )
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_9662_gcd__non__0__int,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Y)
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Y) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Y)) ) ) ).

% gcd_non_0_int
tff(fact_9663_gcd__unique__int,axiom,
    ! [D: int,A2: int,B2: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),A2)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B2)
        & ! [E4: int] :
            ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E4),A2)
              & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E4),B2) )
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E4),D) ) )
    <=> ( D = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A2),B2) ) ) ).

% gcd_unique_int
tff(fact_9664_gcd__code__int,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(gcd_gcd_int,K),L) = aa_int_int(abs_abs_int,
        $ite(L = zero_zero_int,K,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))))) ).

% gcd_code_int
tff(fact_9665_Frct__code__post_I1_J,axiom,
    ! [A2: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),A2)) = zero_zero_rat ).

% Frct_code_post(1)
tff(fact_9666_Frct__code__post_I2_J,axiom,
    ! [A2: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),zero_zero_int)) = zero_zero_rat ).

% Frct_code_post(2)
tff(fact_9667_Frct__code__post_I3_J,axiom,
    frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) = one_one_rat ).

% Frct_code_post(3)
tff(fact_9668_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,L))) = divide_divide_rat(numeral_numeral_rat(K),numeral_numeral_rat(L)) ).

% Frct_code_post(6)
tff(fact_9669_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int)) = numeral_numeral_rat(K) ).

% Frct_code_post(4)
tff(fact_9670_VEBT__internal_Ospace_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),Xb)
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),vEBT_Leaf((A),(B))) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList2),Summary) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList2)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList2),Summary)) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_9671_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B2) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_9672_gcd__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),A2) = A2 ).

% gcd_nat.left_neutral
tff(fact_9673_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B2) )
    <=> ( ( A2 = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_9674_gcd__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),zero_zero_nat) = A2 ).

% gcd_nat.right_neutral
tff(fact_9675_gcd__0__nat,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),zero_zero_nat) = Xb ).

% gcd_0_nat
tff(fact_9676_gcd__0__left__nat,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),Xb) = Xb ).

% gcd_0_left_nat
tff(fact_9677_gcd__1__nat,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_9678_gcd__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_9679_gcd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N))
    <=> ( ( M != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9680_gcd__diff2__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).

% gcd_diff2_nat
tff(fact_9681_gcd__diff1__nat,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) ) ) ).

% gcd_diff1_nat
tff(fact_9682_gcd__le2__nat,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B2)),B2) ) ).

% gcd_le2_nat
tff(fact_9683_gcd__le1__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B2)),A2) ) ).

% gcd_le1_nat
tff(fact_9684_gcd__non__0__nat,axiom,
    ! [Y: nat,Xb: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)) ) ) ).

% gcd_non_0_nat
tff(fact_9685_gcd__nat_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Y) = $ite(Y = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y))) ).

% gcd_nat.simps
tff(fact_9686_gcd__nat_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa))) ) ) ).

% gcd_nat.elims
tff(fact_9687_bezout__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ? [X2: nat,Y2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B2)) ) ).

% bezout_nat
tff(fact_9688_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A2: nat] :
    ? [X2: nat,Y2: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X2))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B2) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),X2))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),X2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9689_gcd__code__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      aa_Cod1783974652533396728nteger(gcd_gcd_Code_integer(K),L) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,
        $ite(L = zero_z3403309356797280102nteger,K,aa_Cod1783974652533396728nteger(gcd_gcd_Code_integer(L),aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L))))) ).

% gcd_code_integer
tff(fact_9690_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),Xb)
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Y = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),vEBT_Leaf((A),(B))) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList2),Summary) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList2)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList2),Summary)) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_9691_VEBT__internal_Ocnt_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: real] :
      ( ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),Xb)
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Y = one_one_real )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),vEBT_Leaf((A),(B))) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList2),Summary) )
               => ( ( Y = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList2)),zero_zero_real)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList2),Summary)) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
tff(fact_9692_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),Xb)
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Y = one_one_nat )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),vEBT_Leaf((A),(B))) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList2),Summary) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList2)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList2),Summary)) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
tff(fact_9693_vebt__maxt_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),Xb)
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Y = $ite(
                      (B),
                      aa_nat_option_nat(some_nat,one_one_nat),
                      $ite((A),aa_nat_option_nat(some_nat,zero_zero_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Leaf((A),(B))) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y = aa_nat_option_nat(some_nat,Ma2) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_9694_vebt__mint_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),Xb)
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Y = $ite(
                      (A),
                      aa_nat_option_nat(some_nat,zero_zero_nat),
                      $ite((B),aa_nat_option_nat(some_nat,one_one_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf((A),(B))) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y = aa_nat_option_nat(some_nat,Mi2) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_9695_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),Xb)
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite((A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),vEBT_Leaf((A),(B))) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw) )
               => ( ( Y = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_9696_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),Xb)
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite((B),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),vEBT_Leaf((A),(B))) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw) )
               => ( ( Y = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_9697_gcd__nat_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9698_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,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( ( Y = one_one_nat )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv)) )
               => ( ( Y = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf($true,(Uv))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ( Y = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2),Uy2) )
                   => ( ( Y = one_one_nat )
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2),Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2) )
                     => ( ( Y = one_one_nat )
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_9699_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
      <=> (Y) )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv)) )
               => ( ~ (Y)
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Y)
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2),Uy2) )
                   => ( (Y)
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2),Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2) )
                     => ( ~ (Y)
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_9700_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ! [Uv: $o] :
              ( ( Xb = vEBT_Leaf($true,(Uv)) )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv))) )
         => ( ! [Uu2: $o] :
                ( ( Xb = vEBT_Leaf((Uu2),$true) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_9701_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
         => ~ ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2),Uy2) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2),Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_9702_setceilmax,axiom,
    ! [S: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,S),M)
     => ( ! [X2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Listy))
           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X2),N) )
       => ( ( M = aa_nat_nat(suc,N) )
         => ( ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Listy))
               => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X2)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,N))) ) )
           => ( ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,S)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) )
             => ( aa_nat_int(semiri1314217659103216013at_int,lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(S,set_VEBT_VEBT2(Listy))))) = aa_real_int(archim7802044766580827645g_real,aa_real_real(log(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_9703_height__compose__list,axiom,
    ! [Tb: vEBT_VEBT,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Tb),set_VEBT_VEBT2(TreeList))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,Tb)),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary2,set_VEBT_VEBT2(TreeList))))) ) ).

% height_compose_list
tff(fact_9704_max__ins__scaled,axiom,
    ! [N: nat,X14: vEBT_VEBT,M: nat,X13: list_VEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),lattic8265883725875713057ax_nat(insert_nat(aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14),image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13))))))) ).

% max_ins_scaled
tff(fact_9705_height__i__max,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,Foo: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,nth_VEBT_VEBT(X13,I))),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Foo),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13))))) ) ).

% height_i_max
tff(fact_9706_max__idx__list,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,N: nat,X14: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_VEBT_VEBT_nat(vEBT_VEBT_height,nth_VEBT_VEBT(X13,I)))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14)),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13)))))))) ) ).

% max_idx_list
tff(fact_9707_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_nc,N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_9708_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,Deg2,TreeList),Summary2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary2,set_VEBT_VEBT2(TreeList))))) ).

% VEBT_internal.height.simps(2)
tff(fact_9709_VEBT__internal_Oheight_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,Xb) = Y )
     => ( ( ? [A: $o,B: $o] : Xb = vEBT_Leaf((A),(B))
         => ( Y != zero_zero_nat ) )
       => ~ ! [Uu2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg,TreeList2),Summary) )
             => ( Y != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList2))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_9710_divide__nat__def,axiom,
    ! [M: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),N) = $ite(N = zero_zero_nat,zero_zero_nat,lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_arh(M),N)))) ).

% divide_nat_def
tff(fact_9711_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),N) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ari(N),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9712_VEBT__internal_Oheight_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),Xb)
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leaf((A),(B)) )
             => ( ( Y = zero_zero_nat )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),vEBT_Leaf((A),(B))) ) )
         => ~ ! [Uu2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg,TreeList2),Summary) )
               => ( ( Y = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList2))))) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg,TreeList2),Summary)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_9713_bij__betw__Suc,axiom,
    ! [M8: set_nat,N2: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M8),N2)
    <=> ( image_nat_nat(suc,M8) = N2 ) ) ).

% bij_betw_Suc
tff(fact_9714_range__mult,axiom,
    ! [A2: real] :
      image_real_real(times_times_real(A2),top_top_set_real) = $ite(A2 = zero_zero_real,insert_real(zero_zero_real,bot_bot_set_real),top_top_set_real) ).

% range_mult
tff(fact_9715_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero_int )
     => ( lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_my,N))) = aa_int_int(abs_abs_int,N) ) ) ).

% Max_divisors_self_int
tff(fact_9716_zero__notin__Suc__image,axiom,
    ! [A3: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),image_nat_nat(suc,A3)) ).

% zero_notin_Suc_image
tff(fact_9717_gcd__is__Max__divisors__int,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M),N) = lattic8263393255366662781ax_int(collect_int(aa_int_fun_int_o(aTP_Lamm_arj(N),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_9718_image__Suc__lessThan,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),N) ).

% image_Suc_lessThan
tff(fact_9719_image__Suc__atMost,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,one_one_nat),aa_nat_nat(suc,N)) ).

% image_Suc_atMost
tff(fact_9720_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9721_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or1269000886237332187st_nat,zero_zero_nat),N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9722_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9723_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,N)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N))) ).

% atMost_Suc_eq_insert_0
tff(fact_9724_range__mod,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ark,N),top_top_set_nat) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),N) ) ) ).

% range_mod
tff(fact_9725_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_arl,L),set_or4662586982721622107an_int(zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L))) = set_or4662586982721622107an_int(L,U) ).

% image_add_int_atLeastLessThan
tff(fact_9726_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : image_4470545334726330049nteger(aTP_Lamm_arm(L),set_or8404916559141939852nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(U),L))) = set_or8404916559141939852nteger(L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_9727_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
     => ( set_or4662586982721622107an_int(zero_zero_int,U) = image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_9728_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Xb: nat,Y: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_arn,C),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,Xb),Y)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y),
        aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9729_UNIV__nat__eq,axiom,
    top_top_set_nat = insert_nat(zero_zero_nat,image_nat_nat(suc,top_top_set_nat)) ).

% UNIV_nat_eq
tff(fact_9730_wait__rule,axiom,
    ! [N: nat] : hoare_8945653483474564448t_unit(one_one_assn,heap_Time_wait(N),aTP_Lamm_aro) ).

% wait_rule
tff(fact_9731_take__bit__num__def,axiom,
    ! [N: nat,M: num] :
      aa_num_option_num(bit_take_bit_num(N),M) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M)) = zero_zero_nat,none_num,aa_num_option_num(some_num,aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),aa_num_nat(numeral_numeral_nat,M))))) ).

% take_bit_num_def
tff(fact_9732_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q2)) = Q2 ).

% num_of_nat_numeral_eq
tff(fact_9733_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9734_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_9735_num__of__nat__One,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),one_one_nat)
     => ( aa_nat_num(num_of_nat,N) = one ) ) ).

% num_of_nat_One
tff(fact_9736_num__of__nat__double,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)) = aa_num_num(bit0,aa_nat_num(num_of_nat,N)) ) ) ).

% num_of_nat_double
tff(fact_9737_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9738_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N),aa_num_num(inc,aa_nat_num(num_of_nat,N)),one) ).

% num_of_nat.simps(2)
tff(fact_9739_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se8568078237143864401it_int(N),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9740_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se8568078237143864401it_int(N),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% drop_bit_negative_int_iff
tff(fact_9741_drop__bit__minus__one,axiom,
    ! [N: nat] : aa_int_int(bit_se8568078237143864401it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% drop_bit_minus_one
tff(fact_9742_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(bit_se8568078237143864401it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9743_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,N = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_9744_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_9745_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(N),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9746_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_9747_drop__bit__int__code_I1_J,axiom,
    ! [I: int] : aa_int_int(bit_se8568078237143864401it_int(zero_zero_nat),I) = I ).

% drop_bit_int_code(1)
tff(fact_9748_drop__bit__int__code_I2_J,axiom,
    ! [N: nat] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,N)),zero_zero_int) = zero_zero_int ).

% drop_bit_int_code(2)
tff(fact_9749_drop__bit__nat__eq,axiom,
    ! [N: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se8568078237143864401it_int(N),K)) ).

% drop_bit_nat_eq
tff(fact_9750_bin__rest__code,axiom,
    ! [I: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,I),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(bit_se8568078237143864401it_int(one_one_nat),I) ).

% bin_rest_code
tff(fact_9751_drop__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(N),K) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% drop_bit_int_def
tff(fact_9752_drop__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),M) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% drop_bit_nat_def
tff(fact_9753_shiftr__integer__conv__div__pow2,axiom,
    ! [N: nat,Xb: code_integer] : aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(N),Xb) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(Xb),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ).

% shiftr_integer_conv_div_pow2
tff(fact_9754_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se545348938243370406it_int(N),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9755_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se545348938243370406it_int(N),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% push_bit_negative_int_iff
tff(fact_9756_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% push_bit_of_Suc_0
tff(fact_9757_drop__bit__push__bit__int,axiom,
    ! [M: nat,N: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(M),aa_int_int(bit_se545348938243370406it_int(N),K)) = aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)),K)) ).

% drop_bit_push_bit_int
tff(fact_9758_flip__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_9759_set__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9760_push__bit__nat__eq,axiom,
    ! [N: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se545348938243370406it_int(N),K)) ).

% push_bit_nat_eq
tff(fact_9761_push__bit__int__code_I1_J,axiom,
    ! [I: int] : aa_int_int(bit_se545348938243370406it_int(zero_zero_nat),I) = I ).

% push_bit_int_code(1)
tff(fact_9762_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_se545348938243370406it_int(M),K)),N)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_9763_shiftl__integer__conv__mult__pow2,axiom,
    ! [N: nat,Xb: code_integer] : aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(N),Xb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),N)) ).

% shiftl_integer_conv_mult_pow2
tff(fact_9764_Bit__Operations_Oset__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se7879613467334960850it_int(N),K) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)) ).

% Bit_Operations.set_bit_int_def
tff(fact_9765_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q2)),N)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9766_lsb__integer__code,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(least_7544222001954398261nteger,Xb)
    <=> aa_nat_o(aa_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,Xb),zero_zero_nat) ) ).

% lsb_integer_code
tff(fact_9767_flip__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se2159334234014336723it_int(N),K) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)) ).

% flip_bit_int_def
tff(fact_9768_unset__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se4203085406695923979it_int(N),K) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(N),one_one_int))) ).

% unset_bit_int_def
tff(fact_9769_push__bit__int__def,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_se545348938243370406it_int(N),K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% push_bit_int_def
tff(fact_9770_push__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ).

% push_bit_nat_def
tff(fact_9771_push__bit__minus__one,axiom,
    ! [N: nat] : aa_int_int(bit_se545348938243370406it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),N)) ).

% push_bit_minus_one
tff(fact_9772_Bit__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: $o] : bits_Bit_integer(aa_int_Code_integer(code_integer_of_int,Xaa),(Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(Xb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa))) ).

% Bit_integer.abs_eq
tff(fact_9773_Bit__integer__code_I1_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,$false) = aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(one_one_nat),I) ).

% Bit_integer_code(1)
tff(fact_9774_Bit__integer__code_I2_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,$true) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(one_one_nat),I)),one_one_Code_integer) ).

% Bit_integer_code(2)
tff(fact_9775_dup__1,axiom,
    aa_Cod1783974652533396728nteger(code_dup,one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% dup_1
tff(fact_9776_bin__last__integer__nbe,axiom,
    ! [I: code_integer] :
      ( aa_Code_integer_o(bits_b8758750999018896077nteger,I)
    <=> ( aa_Cod1783974652533396728nteger(modulo364778990260209775nteger(I),numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) ) ).

% bin_last_integer_nbe
tff(fact_9777_Code__Numeral_Odup__code_I1_J,axiom,
    aa_Cod1783974652533396728nteger(code_dup,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% Code_Numeral.dup_code(1)
tff(fact_9778_bin__last__integer__code,axiom,
    ! [I: code_integer] :
      ( aa_Code_integer_o(bits_b8758750999018896077nteger,I)
    <=> ( aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(I),one_one_Code_integer) != zero_z3403309356797280102nteger ) ) ).

% bin_last_integer_code
tff(fact_9779_bin__last__integer_Oabs__eq,axiom,
    ! [Xb: int] :
      ( aa_Code_integer_o(bits_b8758750999018896077nteger,aa_int_Code_integer(code_integer_of_int,Xb))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb) ) ).

% bin_last_integer.abs_eq
tff(fact_9780_bitAND__integer__unfold,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(Xb),Y) = $ite(
        Xb = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(
          Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),
          Y,
          bits_Bit_integer(aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Xb)),aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Y)),
            ( aa_Code_integer_o(bits_b8758750999018896077nteger,Xb)
            & aa_Code_integer_o(bits_b8758750999018896077nteger,Y) )) ) ) ).

% bitAND_integer_unfold
tff(fact_9781_bitOR__integer__unfold,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(Xb),Y) = $ite(
        Xb = zero_z3403309356797280102nteger,
        Y,
        $ite(
          Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),
          aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),
          bits_Bit_integer(aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Xb)),aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Y)),
            ( aa_Code_integer_o(bits_b8758750999018896077nteger,Xb)
            | aa_Code_integer_o(bits_b8758750999018896077nteger,Y) )) ) ) ).

% bitOR_integer_unfold
tff(fact_9782_bin__rest__integer__code,axiom,
    ! [I: code_integer] : aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,I) = aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(I),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% bin_rest_integer_code
tff(fact_9783_bin__rest__integer_Oabs__eq,axiom,
    ! [Xb: int] : aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% bin_rest_integer.abs_eq
tff(fact_9784_bitXOR__integer__unfold,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      aa_Cod1783974652533396728nteger(bit_se3222712562003087583nteger(Xb),Y) = $ite(
        Xb = zero_z3403309356797280102nteger,
        Y,
        $ite(
          Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),
          aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,Y),
          bits_Bit_integer(aa_Cod1783974652533396728nteger(bit_se3222712562003087583nteger(aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Xb)),aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Y)),
            ~ aa_Code_integer_o(bits_b8758750999018896077nteger,Xb) = aa_Code_integer_o(bits_b8758750999018896077nteger,Y)) ) ) ).

% bitXOR_integer_unfold
tff(fact_9785_Uint32__code,axiom,
    ! [I: code_integer] :
      aa_Cod8401701657498852871uint32(uint322,I) = $let(
        i: code_integer,
        i:= aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(I),numera6620942414471956472nteger(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))))))))))))))))))))))))))))))),
        $ite(aa_nat_o(aa_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,i),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_Cod8401701657498852871uint32(uint32_signed,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(i),numera6620942414471956472nteger(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))))))))))))))))))))),aa_Cod8401701657498852871uint32(uint32_signed,i)) ) ).

% Uint32_code
tff(fact_9786_Uint32__signed__def,axiom,
    ! [I: code_integer] :
      aa_Cod8401701657498852871uint32(uint32_signed,I) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))))))))))))))))))))
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))))))))))))))))))),I) ),
        aa_Cod8401701657498852871uint32(aa_fun8111848626357839434uint32(undefi1969809534184926113uint32,uint322),I),
        aa_Cod8401701657498852871uint32(uint322,I) ) ).

% Uint32_signed_def
tff(fact_9787_vebt__minti_Opelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Xb) = Y )
     => ( aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),Xb)
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leafi((A),(B)) )
             => ( ( Y = $ite(
                      (A),
                      heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),
                      $ite((B),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),heap_T3487192422709364219on_nat(none_nat)) ) )
               => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),vEBT_Leafi((A),(B))) ) )
         => ( ! [Uu2: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( ( Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv),Uw) )
               => ( ( Y = heap_T3487192422709364219on_nat(none_nat) )
                 => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv),Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                  ( ( Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y = heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,Mi2)) )
                   => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2)) ) ) ) ) ) ) ).

% vebt_minti.pelims
tff(fact_9788_vebt__maxti_Opelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Xb) = Y )
     => ( aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),Xb)
       => ( ! [A: $o,B: $o] :
              ( ( Xb = vEBT_Leafi((A),(B)) )
             => ( ( Y = $ite(
                      (B),
                      heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),
                      $ite((A),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)) ) )
               => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),vEBT_Leafi((A),(B))) ) )
         => ( ! [Uu2: nat,Uv: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
                ( ( Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv),Uw) )
               => ( ( Y = heap_T3487192422709364219on_nat(none_nat) )
                 => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv),Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
                  ( ( Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2) )
                 => ( ( Y = heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,Ma2)) )
                   => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),Ux2,Uy2),Uz2)) ) ) ) ) ) ) ).

% vebt_maxti.pelims
tff(fact_9789_VEBT__internal_OminNulli_Opelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_Time_Heap_o] :
      ( ( aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Xb) = Y )
     => ( aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),Xb)
       => ( ( ( Xb = vEBT_Leafi($false,$false) )
           => ( ( Y = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true) )
             => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),vEBT_Leafi($false,$false)) ) )
         => ( ! [Uv: $o] :
                ( ( Xb = vEBT_Leafi($true,(Uv)) )
               => ( ( Y = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) )
                 => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),vEBT_Leafi($true,(Uv))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leafi((Uu2),$true) )
                 => ( ( Y = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) )
                   => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),vEBT_Leafi((Uu2),$true)) ) )
             => ( ! [Uw: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] :
                    ( ( Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw,Ux2),Uy2) )
                   => ( ( Y = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true) )
                     => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw,Ux2),Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] :
                      ( ( Xb = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2) )
                     => ( ( Y = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) )
                       => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz2),Va3,Vb2),Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.pelims
tff(fact_9790_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list_int] :
      upto_aux(I,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),Js,upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js))) ).

% upto_aux_rec
tff(fact_9791_upto_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: list_int] :
      ( ( upto(Xb,Xaa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa),cons_int(Xb,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int),Xaa)),nil_int) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_9792_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I),J))
     => ( upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_9793_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I)
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_9794_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil2
tff(fact_9795_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil
tff(fact_9796_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K))),J)
     => ( aa_nat_int(nth_int(upto(I,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_9797_length__upto,axiom,
    ! [I: int,J: int] : size_size_list_int(upto(I,J)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),I)),one_one_int)) ).

% length_upto
tff(fact_9798_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,N)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)),cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_num_int(numeral_numeral_int,N))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_9799_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_9800_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,N)),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_num_int(numeral_numeral_int,N))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_9801_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_9802_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9803_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9804_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or4662586982721622107an_int(I,J) = set_int2(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_9805_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9806_upto_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: list_int] :
      ( ( upto(Xb,Xaa) = Y )
     => ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa),cons_int(Xb,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int),Xaa)),nil_int) ) ) ).

% upto.elims
tff(fact_9807_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ).

% upto.simps
tff(fact_9808_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9809_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_9810_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(suc,N)),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_9811_concat__bit__0,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(zero_zero_nat),K),L) = L ).

% concat_bit_0
tff(fact_9812_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),zero_zero_int) = aa_int_int(bit_se2923211474154528505it_int(N),K) ).

% concat_bit_of_zero_2
tff(fact_9813_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_9814_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ).

% concat_bit_negative_iff
tff(fact_9815_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),zero_zero_int),L) = aa_int_int(bit_se545348938243370406it_int(N),L) ).

% concat_bit_of_zero_1
tff(fact_9816_concat__bit__take__bit__eq,axiom,
    ! [N: nat,B2: int] : aa_int_fun_int_int(bit_concat_bit(N),aa_int_int(bit_se2923211474154528505it_int(N),B2)) = aa_int_fun_int_int(bit_concat_bit(N),B2) ).

% concat_bit_take_bit_eq
tff(fact_9817_concat__bit__eq__iff,axiom,
    ! [N: nat,K: int,L: int,R: int,S: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),R),S) )
    <=> ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_int_int(bit_se2923211474154528505it_int(N),R) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
tff(fact_9818_concat__bit__assoc,axiom,
    ! [N: nat,K: int,M: nat,L: int,R: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),L),R)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)),R) ).

% concat_bit_assoc
tff(fact_9819_concat__bit__eq,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(bit_se545348938243370406it_int(N),L)) ).

% concat_bit_eq
tff(fact_9820_concat__bit__def,axiom,
    ! [N: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(bit_se2923211474154528505it_int(N),K)),aa_int_int(bit_se545348938243370406it_int(N),L)) ).

% concat_bit_def
tff(fact_9821_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),K),L)),N)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_9822_signed__take__bit__eq__concat__bit,axiom,
    ! [N: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(N),K) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),aa_int_int(uminus_uminus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_9823_int__set__bit__conv__ops,axiom,
    ! [I: int,N: nat,B2: $o] :
      aa_o_int(generi8991105624351003935it_int(I,N),(B2)) = $ite((B2),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,I),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,I),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(N),one_one_int)))) ).

% int_set_bit_conv_ops
tff(fact_9824_int__set__bit__True__conv__OR,axiom,
    ! [I: int,N: nat] : aa_o_int(generi8991105624351003935it_int(I,N),$true) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,I),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)) ).

% int_set_bit_True_conv_OR
tff(fact_9825_int__set__bit__False__conv__NAND,axiom,
    ! [I: int,N: nat] : aa_o_int(generi8991105624351003935it_int(I,N),$false) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,I),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(N),one_one_int))) ).

% int_set_bit_False_conv_NAND
tff(fact_9826_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_o] : aa_int_o(aa_int_fun_int_o(ord_less_int,groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),size_size_list_o(Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_9827_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( upt(zero_zero_nat,J) = nil_nat )
    <=> ( J = zero_zero_nat ) ) ).

% upt_0_eq_Nil_conv
tff(fact_9828_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_9829_upt__merge,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K) )
     => ( append_nat(upt(I,J),upt(J,K)) = upt(I,K) ) ) ).

% upt_merge
tff(fact_9830_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_9831_length__upt,axiom,
    ! [I: nat,J: nat] : size_size_list_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) ).

% length_upt
tff(fact_9832_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N)),cons_nat(aa_num_nat(numeral_numeral_nat,M),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,N))),nil_nat) ).

% upt_rec_numeral
tff(fact_9833_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)
     => ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_9834_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J),cons_nat(I,upt(aa_nat_nat(suc,I),J)),nil_nat) ).

% upt_rec
tff(fact_9835_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_9836_map__add__upt_H,axiom,
    ! [Ofs: nat,A2: nat,B2: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_arp,Ofs),upt(A2,B2)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),Ofs),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),Ofs)) ).

% map_add_upt'
tff(fact_9837_map__add__upt,axiom,
    ! [N: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_arp,N),upt(zero_zero_nat,M)) = upt(N,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),N)) ).

% map_add_upt
tff(fact_9838_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_9839_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_9840_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs: list_nat,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( upt(I,J) = append_nat(Xs,Ys) )
      <=> ? [K3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),J)
            & ( upt(I,K3) = Xs )
            & ( upt(K3,J) = Ys ) ) ) ) ).

% upt_eq_append_conv
tff(fact_9841_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( append_nat(upt(zero_zero_nat,I),upt(I,J)) = upt(zero_zero_nat,J) ) ) ).

% upt_append
tff(fact_9842_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_9843_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      upt(I,aa_nat_nat(suc,J)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J),append_nat(upt(I,J),cons_nat(J,nil_nat)),nil_nat) ).

% upt_Suc
tff(fact_9844_upt__eq__lel__conv,axiom,
    ! [L: nat,H: nat,Is1: list_nat,I: nat,Is2: list_nat] :
      ( ( upt(L,H) = append_nat(Is1,cons_nat(I,Is2)) )
    <=> ( ( Is1 = upt(L,I) )
        & ( Is2 = upt(aa_nat_nat(suc,I),H) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),H) ) ) ).

% upt_eq_lel_conv
tff(fact_9845_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xb: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(Xb,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( I = Xb )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9846_map__decr__upt,axiom,
    ! [M: nat,N: nat] : map_nat_nat(aTP_Lamm_arq,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_9847_atLeast__upt,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = set_nat2(upt(zero_zero_nat,N)) ).

% atLeast_upt
tff(fact_9848_atMost__upto,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,N))) ).

% atMost_upto
tff(fact_9849_map__bit__range__eq__if__take__bit__eq,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),K) = aa_int_int(bit_se2923211474154528505it_int(N),L) )
     => ( map_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),upt(zero_zero_nat,N)) = map_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),upt(zero_zero_nat,N)) ) ) ).

% map_bit_range_eq_if_take_bit_eq
tff(fact_9850_int__sdiv__simps_I2_J,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A2),zero_zero_int) = zero_zero_int ).

% int_sdiv_simps(2)
tff(fact_9851_sdiv__int__0__div,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,zero_zero_int),Xb) = zero_zero_int ).

% sdiv_int_0_div
tff(fact_9852_sdiv__int__div__0,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,Xb),zero_zero_int) = zero_zero_int ).

% sdiv_int_div_0
tff(fact_9853_int__sdiv__simps_I1_J,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A2),one_one_int) = A2 ).

% int_sdiv_simps(1)
tff(fact_9854_int__sdiv__same__is__1,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A2),B2) = A2 )
      <=> ( B2 = one_one_int ) ) ) ).

% int_sdiv_same_is_1
tff(fact_9855_int__sdiv__simps_I3_J,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A2),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A2) ).

% int_sdiv_simps(3)
tff(fact_9856_sdiv__int__numeral__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) ).

% sdiv_int_numeral_numeral
tff(fact_9857_int__sdiv__negated__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A2),B2) = aa_int_int(uminus_uminus_int,A2) )
      <=> ( B2 = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% int_sdiv_negated_is_minus1
tff(fact_9858_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A2),B2) != zero_zero_int )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,A2),B2)) = aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)) ) ) ).

% sgn_sdiv_eq_sgn_mult
tff(fact_9859_signed__divide__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(signed6714573509424544716de_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ).

% signed_divide_int_def
tff(fact_9860_upt__filter__extend,axiom,
    ! [U: nat,U2: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),U2)
     => ( ! [I2: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,U),I2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),U2) )
           => ~ aa_nat_o(P,I2) )
       => ( filter_nat2(P,upt(zero_zero_nat,U)) = filter_nat2(P,upt(zero_zero_nat,U2)) ) ) ) ).

% upt_filter_extend
tff(fact_9861_sort__upt,axiom,
    ! [M: nat,N: nat] : linord738340561235409698at_nat(aTP_Lamm_eb,upt(M,N)) = upt(M,N) ).

% sort_upt
tff(fact_9862_sort__upto,axiom,
    ! [I: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_ja,upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_9863_fails__assert_H,axiom,
    ! [P: $o,H: heap_e7401611519738050253t_unit] :
      ( time_f8834461667527620124t_unit(refine_Imp_assert((P)),H)
    <=> ~ (P) ) ).

% fails_assert'
tff(fact_9864_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( hd_nat(upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_9865_Cauchy__iff2,axiom,
    ! [X5: fun_nat_real] :
      ( topolo4055970368930404560y_real(X5)
    <=> ! [J3: nat] :
        ? [M9: nat] :
        ! [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),M6)
         => ! [N8: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),N8)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(X5,M6)),aa_nat_real(X5,N8)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_9866_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(size_size_VEBT_VEBT,X13)),aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size(3)
tff(fact_9867_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(vEBT_size_VEBT,X13)),aa_VEBT_VEBT_nat(vEBT_size_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size_gen(1)
tff(fact_9868_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_9869_smod__int__range,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2)),set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,B2))),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,B2)),one_one_int))) ) ).

% smod_int_range
tff(fact_9870_smod__int__0__mod,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,zero_zero_int),Xb) = zero_zero_int ).

% smod_int_0_mod
tff(fact_9871_smod__int__mod__0,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,Xb),zero_zero_int) = Xb ).

% smod_int_mod_0
tff(fact_9872_smod__int__numeral__numeral,axiom,
    ! [M: num,N: num] : aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,N)) ).

% smod_int_numeral_numeral
tff(fact_9873_smod__int__compares_I8_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2)) ) ) ).

% smod_int_compares(8)
tff(fact_9874_smod__int__compares_I7_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2)),zero_zero_int) ) ) ).

% smod_int_compares(7)
tff(fact_9875_smod__int__compares_I6_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2)) ) ) ).

% smod_int_compares(6)
tff(fact_9876_smod__int__compares_I4_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2)),zero_zero_int) ) ) ).

% smod_int_compares(4)
tff(fact_9877_smod__int__compares_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2)) ) ) ).

% smod_int_compares(2)
tff(fact_9878_smod__int__compares_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2)),B2) ) ) ).

% smod_int_compares(1)
tff(fact_9879_smod__mod__positive,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => ( aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) ) ) ) ).

% smod_mod_positive
tff(fact_9880_smod__int__compares_I5_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2)),aa_int_int(uminus_uminus_int,B2)) ) ) ).

% smod_int_compares(5)
tff(fact_9881_smod__int__compares_I3_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(aa_int_fun_int_int(signed6292675348222524329lo_int,A2),B2)) ) ) ).

% smod_int_compares(3)
tff(fact_9882_valid__eq2,axiom,
    ! [Tb: vEBT_VEBT,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Tb),D)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),D) ) ).

% valid_eq2
tff(fact_9883_valid__eq,axiom,
    ! [Tb: vEBT_VEBT,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Tb),D)
    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),D) ) ).

% valid_eq
tff(fact_9884_valid__eq1,axiom,
    ! [Tb: vEBT_VEBT,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tb),D)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Tb),D) ) ).

% valid_eq1
tff(fact_9885_uint32_Osize__eq__length,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))) = type_l796852477590012082l_num1(type_N8448461349408098053l_num1) ).

% uint32.size_eq_length
tff(fact_9886_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o,D: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,vEBT_Leaf((Uu),(Uv2))),D)
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_9887_len__num0,axiom,
    ! [Uu: itself_Numeral_num0] : type_l4264026598287037464l_num0(Uu) = zero_zero_nat ).

% len_num0
tff(fact_9888_len__num1,axiom,
    ! [Uu: itself_Numeral_num1] : type_l4264026598287037465l_num1(Uu) = one_one_nat ).

% len_num1
tff(fact_9889_len__of__finite__1__def,axiom,
    ! [Xb: itself_finite_1] : type_l31302759751748491nite_1(Xb) = one_one_nat ).

% len_of_finite_1_def
tff(fact_9890_len__of__finite__3__def,axiom,
    ! [Xb: itself_finite_3] : type_l31302759751748493nite_3(Xb) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% len_of_finite_3_def
tff(fact_9891_len__of__finite__2__def,axiom,
    ! [Xb: itself_finite_2] : type_l31302759751748492nite_2(Xb) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% len_of_finite_2_def
tff(fact_9892_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)) ).

% min_Suc_Suc
tff(fact_9893_min__0L,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N) = zero_zero_nat ).

% min_0L
tff(fact_9894_min__0R,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9895_min__Suc__gt_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,A2)),B2) = aa_nat_nat(suc,A2) ) ) ).

% min_Suc_gt(1)
tff(fact_9896_min__Suc__gt_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B2),aa_nat_nat(suc,A2)) = aa_nat_nat(suc,A2) ) ) ).

% min_Suc_gt(2)
tff(fact_9897_min__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_9898_min__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,N)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),N)) ).

% min_numeral_Suc
tff(fact_9899_concat__bit__assoc__sym,axiom,
    ! [M: nat,N: nat,K: int,L: int,R: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(M),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)),R) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),K),aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),L),R)) ).

% concat_bit_assoc_sym
tff(fact_9900_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),I) ).

% min_diff
tff(fact_9901_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,N),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)) ).

% nat_mult_min_right
tff(fact_9902_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),Q2)) ).

% nat_mult_min_left
tff(fact_9903_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(M),aa_int_int(aa_int_fun_int_int(bit_concat_bit(N),K),L)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N)),K),aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),N)),L)) ).

% take_bit_concat_bit_eq
tff(fact_9904_min__Suc2,axiom,
    ! [M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,N)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_arr,N)),M) ).

% min_Suc2
tff(fact_9905_min__Suc1,axiom,
    ! [N: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,N)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_ars,N)),M) ).

% min_Suc1
tff(fact_9906_mod__mod__power,axiom,
    ! [K: nat,M: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),N))) ).

% mod_mod_power
tff(fact_9907_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : ord_mi8085742599997312461d_enat(zero_z5237406670263579293d_enat,Q2) = zero_z5237406670263579293d_enat ).

% min_enat_simps(3)
tff(fact_9908_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : ord_mi8085742599997312461d_enat(Q2,zero_z5237406670263579293d_enat) = zero_z5237406670263579293d_enat ).

% min_enat_simps(2)
tff(fact_9909_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
     => ( ( ? [Uu2: $o,Uv: $o] : Xb = vEBT_Leaf((Uu2),(Uv))
         => ( Xaa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList2),Summary) )
             => ( ( Deg = Xaa )
                & $let(
                    n: nat,
                    n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                    $let(
                      m: nat,
                      m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                      ( ! [X: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
                         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X),n) )
                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m)
                      & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                      & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                            & ! [X: vEBT_VEBT] :
                                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
                               => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_art(Deg),TreeList2),Summary),n),m))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9910_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
     => ( ( ? [Uu2: $o,Uv: $o] : Xb = vEBT_Leaf((Uu2),(Uv))
         => ( Xaa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList2),Summary) )
             => ~ ( ( Deg = Xaa )
                  & $let(
                      n: nat,
                      n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $let(
                        m: nat,
                        m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                        ( ! [X: vEBT_VEBT] :
                            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
                           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X),n) )
                        & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m)
                        & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                        & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                              & ! [X: vEBT_VEBT] :
                                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
                                 => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_art(Deg),TreeList2),Summary),n),m))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9911_int__set__bits__K__False,axiom,
    bit_bi6516823479961619367ts_int(aTP_Lamm_fs) = zero_zero_int ).

% int_set_bits_K_False
tff(fact_9912_int__set__bits__K__True,axiom,
    bit_bi6516823479961619367ts_int(aTP_Lamm_fn) = aa_int_int(uminus_uminus_int,one_one_int) ).

% int_set_bits_K_True
tff(fact_9913_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg4: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima2,Deg2,TreeList),Summary2)),Deg4)
    <=> ( ( Deg2 = Deg4 )
        & $let(
            n: nat,
            n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            $let(
              m: nat,
              m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n),
              ( ! [X: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X),n) )
              & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary2),m)
              & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
              & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X6)
                    & ! [X: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList))
                       => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_art(Deg2),TreeList),Summary2),n),m))),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9914_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv: $o] : Xb = vEBT_Leaf((Uu2),(Uv))
         => ( (Y)
          <=> ( Xaa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList2),Summary) )
             => ( (Y)
              <=> ~ ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m: nat,
                          m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X: vEBT_VEBT] :
                              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
                             => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X),n) )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m)
                          & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                                & ! [X: vEBT_VEBT] :
                                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_art(Deg),TreeList2),Summary),n),m))),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9915_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv)) )
             => ( ( (Y)
                <=> ( Xaa = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv))),Xaa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList2),Summary) )
               => ( ( (Y)
                  <=> ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m: nat,
                            m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X: vEBT_VEBT] :
                                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
                               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X),n) )
                            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m)
                            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                                  & ! [X: vEBT_VEBT] :
                                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
                                     => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_art(Deg),TreeList2),Summary),n),m))),Mima) ) ) ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList2),Summary)),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9916_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv))),Xaa))
               => ( Xaa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList2),Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList2),Summary)),Xaa))
                 => ~ ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m: nat,
                            m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X: vEBT_VEBT] :
                                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
                               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X),n) )
                            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m)
                            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                                  & ! [X: vEBT_VEBT] :
                                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
                                     => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_art(Deg),TreeList2),Summary),n),m))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9917_fi__match__entails,axiom,
    ! [M: list_P8527749157015355191n_assn] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( member7957490590177025114n_assn(X2,set_Pr1139785259514867910n_assn(M))
         => aa_Pro2491721552408928739assn_o(produc7274209992780475162assn_o(entails),X2) )
     => aa_assn_o(aa_assn_fun_assn_o(entails,foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,M),one_one_assn)),foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,M),one_one_assn)) ) ).

% fi_match_entails
tff(fact_9918_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(Xb),Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv))),Xaa))
               => ( Xaa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList2),Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,aa_nat4363559699391081921BT_nat(produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList2),Summary)),Xaa))
                 => ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m: nat,
                          m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X: vEBT_VEBT] :
                              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
                             => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,X),n) )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Summary),m)
                          & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X6)
                                & ! [X: vEBT_VEBT] :
                                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_art(Deg),TreeList2),Summary),n),m))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9919_set__bits__int__def,axiom,
    ! [Fa: fun_nat_o] :
      bit_bi6516823479961619367ts_int(Fa) = $ite(
        ? [N8: nat] :
        ! [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),M6)
         => ( aa_nat_o(Fa,M6) = aa_nat_o(Fa,N8) ) ),
        $let(
          n: nat,
          n:= ord_Least_nat(aTP_Lamm_aru(Fa)),
          aa_int_int(bit_ri631733984087533419it_int(n),groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),map_nat_o(Fa,upt(zero_zero_nat,aa_nat_nat(suc,n))))) ),
        zero_zero_int ) ).

% set_bits_int_def
tff(fact_9920_set__bits__int__unfold_H,axiom,
    ! [Fa: fun_nat_o] :
      bit_bi6516823479961619367ts_int(Fa) = $ite(
        ? [N8: nat] :
        ! [N11: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N11)
         => ~ aa_nat_o(Fa,N11) ),
        groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),map_nat_o(Fa,upt(zero_zero_nat,ord_Least_nat(aTP_Lamm_arv(Fa))))),
        $ite(
          ? [N8: nat] :
          ! [N11: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N11)
           => aa_nat_o(Fa,N11) ),
          $let(
            n: nat,
            n:= ord_Least_nat(aTP_Lamm_arw(Fa)),
            aa_int_int(bit_ri631733984087533419it_int(n),groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),append_o(map_nat_o(Fa,upt(zero_zero_nat,n)),cons_o($true,nil_o)))) ),
          zero_zero_int ) ) ).

% set_bits_int_unfold'
tff(fact_9921_Least__eq__0,axiom,
    ! [P: fun_nat_o] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ord_Least_nat(P) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_9922_Least__Suc,axiom,
    ! [P: fun_nat_o,N: nat] :
      ( aa_nat_o(P,N)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_arx(P))) ) ) ) ).

% Least_Suc
tff(fact_9923_Least__Suc2,axiom,
    ! [P: fun_nat_o,N: nat,Q: fun_nat_o,M: nat] :
      ( aa_nat_o(P,N)
     => ( aa_nat_o(Q,M)
       => ( ~ aa_nat_o(P,zero_zero_nat)
         => ( ! [K2: nat] :
                ( aa_nat_o(P,aa_nat_nat(suc,K2))
              <=> aa_nat_o(Q,K2) )
           => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_9924_FI__RESULT__def,axiom,
    ! [M8: list_P8527749157015355191n_assn,UP: assn,UQ: assn,F4: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(fI_RESULT(M8,UP),UQ),F4)
    <=> ( ! [X: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(M8))
           => aa_Pro2491721552408928739assn_o(produc7274209992780475162assn_o(entails),X) )
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,M8),one_one_assn)),UP)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,M8),one_one_assn)),UQ)),F4)) ) ) ).

% FI_RESULT_def
tff(fact_9925_FI__def,axiom,
    ! [M: list_P8527749157015355191n_assn,P2: assn,Q2: assn,Up: assn,Uq: assn,Fa: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(fi(M,P2,Q2,Up),Uq),Fa)
    <=> ( ! [X: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(X,set_Pr1139785259514867910n_assn(M))
           => aa_Pro2491721552408928739assn_o(produc7274209992780475162assn_o(entails),X) )
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc9167289414957590229n_assn,M),one_one_assn)),P2)),Up)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,foldr_assn_assn(times_times_assn,map_Pr8991440229025900053n_assn(produc2051961928117032727n_assn,M),one_one_assn)),Q2)),Uq)),Fa)) ) ) ).

% FI_def
tff(fact_9926_bin__last__set__bits,axiom,
    ! [Fa: fun_nat_o] :
      ( bit_wf_set_bits_int(Fa)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_bi6516823479961619367ts_int(Fa))
      <=> aa_nat_o(Fa,zero_zero_nat) ) ) ).

% bin_last_set_bits
tff(fact_9927_wf__set__bits__int__Suc,axiom,
    ! [Fa: fun_nat_o] :
      ( bit_wf_set_bits_int(aTP_Lamm_arx(Fa))
    <=> bit_wf_set_bits_int(Fa) ) ).

% wf_set_bits_int_Suc
tff(fact_9928_wf__set__bits__int__simps,axiom,
    ! [Fa: fun_nat_o] :
      ( bit_wf_set_bits_int(Fa)
    <=> ? [N8: nat] :
          ( ! [N11: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N11)
             => ~ aa_nat_o(Fa,N11) )
          | ! [N11: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N11)
             => aa_nat_o(Fa,N11) ) ) ) ).

% wf_set_bits_int_simps
tff(fact_9929_zeros,axiom,
    ! [N: nat,Fa: fun_nat_o] :
      ( ! [N7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N7)
         => ~ aa_nat_o(Fa,N7) )
     => bit_wf_set_bits_int(Fa) ) ).

% zeros
tff(fact_9930_wf__set__bits__int_Osimps,axiom,
    ! [Fa: fun_nat_o] :
      ( bit_wf_set_bits_int(Fa)
    <=> ( ? [N8: nat] :
          ! [N11: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N11)
           => ~ aa_nat_o(Fa,N11) )
        | ? [N8: nat] :
          ! [N11: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N11)
           => aa_nat_o(Fa,N11) ) ) ) ).

% wf_set_bits_int.simps
tff(fact_9931_wf__set__bits__int_Ocases,axiom,
    ! [Fa: fun_nat_o] :
      ( bit_wf_set_bits_int(Fa)
     => ( ! [N3: nat] :
            ~ ! [N6: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N3),N6)
               => ~ aa_nat_o(Fa,N6) )
       => ~ ! [N3: nat] :
              ~ ! [N6: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N3),N6)
                 => aa_nat_o(Fa,N6) ) ) ) ).

% wf_set_bits_int.cases
tff(fact_9932_ones,axiom,
    ! [N: nat,Fa: fun_nat_o] :
      ( ! [N7: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N7)
         => aa_nat_o(Fa,N7) )
     => bit_wf_set_bits_int(Fa) ) ).

% ones
tff(fact_9933_wf__set__bits__int__const,axiom,
    ! [B2: $o] : bit_wf_set_bits_int(aa_o_fun_nat_o(aTP_Lamm_agd,(B2))) ).

% wf_set_bits_int_const
tff(fact_9934_int__set__bits__unfold__BIT,axiom,
    ! [Fa: fun_nat_o] :
      ( bit_wf_set_bits_int(Fa)
     => ( bit_bi6516823479961619367ts_int(Fa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(Fa,zero_zero_nat))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),bit_bi6516823479961619367ts_int(comp_nat_o_nat(Fa,suc)))) ) ) ).

% int_set_bits_unfold_BIT
tff(fact_9935_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,C5: $o,C6: $o,C7: $o] :
      ( ord_less_eq_char(aa_o_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62)),(B72)),aa_o_char(char2((C0),(C1),(C22),(C32),(C42),(C5),(C6)),(C7)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_o_nat(aTP_Lamm_ary,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)),aa_nat_nat(foldr_o_nat(aTP_Lamm_ary,cons_o((C0),cons_o((C1),cons_o((C22),cons_o((C32),cons_o((C42),cons_o((C5),cons_o((C6),cons_o((C7),nil_o))))))))),zero_zero_nat)) ) ).

% less_eq_char_simp
tff(fact_9936_bin__rest__set__bits,axiom,
    ! [Fa: fun_nat_o] :
      ( bit_wf_set_bits_int(Fa)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,bit_bi6516823479961619367ts_int(Fa)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = bit_bi6516823479961619367ts_int(comp_nat_o_nat(Fa,suc)) ) ) ).

% bin_rest_set_bits
tff(fact_9937_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,C5: $o,C6: $o,C7: $o] :
      ( ord_less_char(aa_o_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62)),(B72)),aa_o_char(char2((C0),(C1),(C22),(C32),(C42),(C5),(C6)),(C7)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(foldr_o_nat(aTP_Lamm_ary,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)),aa_nat_nat(foldr_o_nat(aTP_Lamm_ary,cons_o((C0),cons_o((C1),cons_o((C22),cons_o((C32),cons_o((C42),cons_o((C5),cons_o((C6),cons_o((C7),nil_o))))))))),zero_zero_nat)) ) ).

% less_char_simp
tff(fact_9938_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(aa_o_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62)),(B72))) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((B72))),numera6620942414471956472nteger(aa_num_num(bit0,one)))),zero_n356916108424825756nteger((B62)))),numera6620942414471956472nteger(aa_num_num(bit0,one)))),zero_n356916108424825756nteger((B52)))),numera6620942414471956472nteger(aa_num_num(bit0,one)))),zero_n356916108424825756nteger((B42)))),numera6620942414471956472nteger(aa_num_num(bit0,one)))),zero_n356916108424825756nteger((B32)))),numera6620942414471956472nteger(aa_num_num(bit0,one)))),zero_n356916108424825756nteger((B22)))),numera6620942414471956472nteger(aa_num_num(bit0,one)))),zero_n356916108424825756nteger((B1)))),numera6620942414471956472nteger(aa_num_num(bit0,one)))),zero_n356916108424825756nteger((B0))) ).

% integer_of_char_code
tff(fact_9939_less__char__def,axiom,
    ! [C1: char,C22: char] :
      ( ord_less_char(C1,C22)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C1)),aa_char_nat(comm_s629917340098488124ar_nat,C22)) ) ).

% less_char_def
tff(fact_9940_nat__of__char__less__256,axiom,
    ! [C: char] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% nat_of_char_less_256
tff(fact_9941_less__eq__char__def,axiom,
    ! [C1: char,C22: char] :
      ( ord_less_eq_char(C1,C22)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_char_nat(comm_s629917340098488124ar_nat,C1)),aa_char_nat(comm_s629917340098488124ar_nat,C22)) ) ).

% less_eq_char_def
tff(fact_9942_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% range_nat_of_char
tff(fact_9943_char_Osize_I2_J,axiom,
    ! [X1: $o,X22: $o,X33: $o,X42: $o,X52: $o,X62: $o,X7: $o,X8: $o] : size_size_char(aa_o_char(char2((X1),(X22),(X33),(X42),(X52),(X62),(X7)),(X8))) = zero_zero_nat ).

% char.size(2)
tff(fact_9944_char_Osize__gen,axiom,
    ! [X1: $o,X22: $o,X33: $o,X42: $o,X52: $o,X62: $o,X7: $o,X8: $o] : size_char(aa_o_char(char2((X1),(X22),(X33),(X42),(X52),(X62),(X7)),(X8))) = zero_zero_nat ).

% char.size_gen
tff(fact_9945_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% UNIV_char_of_nat
tff(fact_9946_Code__Target__Int_Onegative__def,axiom,
    code_Target_negative = comp_int_int_num(uminus_uminus_int,numeral_numeral_int) ).

% Code_Target_Int.negative_def
tff(fact_9947_String_Ochar__of__ascii__of,axiom,
    ! [C: char] : aa_char_nat(comm_s629917340098488124ar_nat,ascii_of(C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_char_nat(comm_s629917340098488124ar_nat,C)) ).

% String.char_of_ascii_of
tff(fact_9948_shiftl__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_Sh3965577149348748681tl_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N) ).

% shiftl_Suc_0
tff(fact_9949_shiftr__Suc__0,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_Sh2154871086232339855tr_nat,aa_nat_nat(suc,zero_zero_nat)),N) = aa_o_nat(zero_n2687167440665602831ol_nat,N = zero_zero_nat) ).

% shiftr_Suc_0
tff(fact_9950_msb__numeral_I1_J,axiom,
    ! [N: num] : ~ aa_int_o(most_s5051101344085556sb_int,aa_num_int(numeral_numeral_int,N)) ).

% msb_numeral(1)
tff(fact_9951_msb__0,axiom,
    ~ aa_int_o(most_s5051101344085556sb_int,zero_zero_int) ).

% msb_0
tff(fact_9952_msb__1,axiom,
    ~ aa_int_o(most_s5051101344085556sb_int,one_one_int) ).

% msb_1
tff(fact_9953_msb__numeral_I2_J,axiom,
    ! [N: num] : aa_int_o(most_s5051101344085556sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,N))) ).

% msb_numeral(2)
tff(fact_9954_msb__bin__rest,axiom,
    ! [Xb: int] :
      ( aa_int_o(most_s5051101344085556sb_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(most_s5051101344085556sb_int,Xb) ) ).

% msb_bin_rest
tff(fact_9955_msb__int__def,axiom,
    ! [Xb: int] :
      ( aa_int_o(most_s5051101344085556sb_int,Xb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int) ) ).

% msb_int_def
tff(fact_9956_msb__integer__code,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(most_s5661112943643946085nteger,Xb)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),zero_z3403309356797280102nteger) ) ).

% msb_integer_code
tff(fact_9957_uint32__msb__test__bit,axiom,
    ! [Xb: uint32] :
      ( aa_uint32_o(most_s9063628576841037300uint32,Xb)
    <=> aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))) ) ).

% uint32_msb_test_bit
tff(fact_9958_msb__uint32__code,axiom,
    ! [Xb: uint32] :
      ( aa_uint32_o(most_s9063628576841037300uint32,Xb)
    <=> aa_Code_integer_o(uint32_test_bit(Xb),numera6620942414471956472nteger(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))) ) ).

% msb_uint32_code
tff(fact_9959_uint32__test__bit__def,axiom,
    ! [Xb: uint32,N: code_integer] :
      ( aa_Code_integer_o(uint32_test_bit(Xb),N)
    <=> $ite(
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,N),zero_z3403309356797280102nteger)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),N) ),
          aa_Code_integer_o(aa_uin5206268876086501546eger_o(aa_fun5262225984791465967eger_o(undefi9030187962712549518eger_o,bit_se5367290876889521763uint32),Xb),N),
          aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Xb),aa_Code_integer_nat(code_nat_of_integer,N)) ) ) ).

% uint32_test_bit_def
tff(fact_9960_test__bit__uint32__code,axiom,
    ! [Xb: uint32,N: nat] :
      ( aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Xb),N)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))
        & aa_Code_integer_o(uint32_test_bit(Xb),code_integer_of_nat(N)) ) ) ).

% test_bit_uint32_code
tff(fact_9961_of__nat__of__integer,axiom,
    ! [K: code_integer] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_Code_integer_nat(code_nat_of_integer,K)) = aa_Cod1783974652533396728nteger(ord_max_Code_integer(zero_z3403309356797280102nteger),K) ).

% of_nat_of_integer
tff(fact_9962_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : aa_Code_integer_nat(code_nat_of_integer,numera6620942414471956472nteger(K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_of_integer_code_post(3)
tff(fact_9963_nat__of__integer__numeral,axiom,
    ! [N: num] : aa_Code_integer_nat(code_nat_of_integer,numera6620942414471956472nteger(N)) = aa_num_nat(numeral_numeral_nat,N) ).

% nat_of_integer_numeral
tff(fact_9964_nat__of__integer__1,axiom,
    aa_Code_integer_nat(code_nat_of_integer,one_one_Code_integer) = one_one_nat ).

% nat_of_integer_1
tff(fact_9965_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_nat(code_nat_of_integer,K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_9966_nat__of__integer__code__post_I1_J,axiom,
    aa_Code_integer_nat(code_nat_of_integer,zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_9967_integer__of__nat__0,axiom,
    code_integer_of_nat(zero_zero_nat) = zero_z3403309356797280102nteger ).

% integer_of_nat_0
tff(fact_9968_integer__of__nat__numeral,axiom,
    ! [N: num] : code_integer_of_nat(aa_num_nat(numeral_numeral_nat,N)) = numera6620942414471956472nteger(N) ).

% integer_of_nat_numeral
tff(fact_9969_integer__of__nat__1,axiom,
    code_integer_of_nat(one_one_nat) = one_one_Code_integer ).

% integer_of_nat_1
tff(fact_9970_integer__of__nat__less__0__conv,axiom,
    ! [N: nat] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,code_integer_of_nat(N)),zero_z3403309356797280102nteger) ).

% integer_of_nat_less_0_conv
tff(fact_9971_nat__of__integer__less__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(code_nat_of_integer,Xb)),aa_Code_integer_nat(code_nat_of_integer,Y))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),Y) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_9972_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U)
     => ( set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U) = image_1215581382706833972nteger(semiri4939895301339042750nteger,aa_nat_set_nat(set_ord_lessThan_nat,aa_Code_integer_nat(code_nat_of_integer,U))) ) ) ).

% image_atLeastZeroLessThan_integer
tff(fact_9973_integer__set__bit__code,axiom,
    ! [Xb: code_integer,N: code_integer,B2: $o] :
      bits_integer_set_bit(Xb,N,(B2)) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,N),zero_z3403309356797280102nteger),
        aa_o_Code_integer(aa_Cod4043041029439962285nteger(aa_Cod3196565106923072233nteger(undefi7543901990407023336nteger,Xb),N),(B2)),
        $ite((B2),aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(Xb),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(aa_Code_integer_nat(code_nat_of_integer,N)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(bit_se3949692690581998587nteger(Xb),aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(aa_Code_integer_nat(code_nat_of_integer,N)),one_one_Code_integer)))) ) ).

% integer_set_bit_code
tff(fact_9974_uint32__shiftr__def,axiom,
    ! [Xb: uint32,N: code_integer] :
      aa_Cod8401701657498852871uint32(uint32_shiftr(Xb),N) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,N),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),N) ),
        aa_Cod8401701657498852871uint32(aa_uin6552852722274082928uint32(aa_fun8838986389337097743uint32(undefi4150998591671624934uint32,bit_se3964402333458159761uint32),Xb),N),
        aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se3964402333458159761uint32,aa_Code_integer_nat(code_nat_of_integer,N)),Xb) ) ).

% uint32_shiftr_def
tff(fact_9975_integer__set__bit__def,axiom,
    ! [Xb: code_integer,N: code_integer,B2: $o] :
      bits_integer_set_bit(Xb,N,(B2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,N),zero_z3403309356797280102nteger),aa_o_Code_integer(aa_Cod4043041029439962285nteger(aa_Cod3196565106923072233nteger(undefi7543901990407023336nteger,Xb),N),(B2)),generi2397576812484419408nteger(Xb,aa_Code_integer_nat(code_nat_of_integer,N),(B2))) ).

% integer_set_bit_def
tff(fact_9976_shiftr__uint32__code,axiom,
    ! [N: nat,Xb: uint32] :
      aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se3964402333458159761uint32,N),Xb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),aa_Cod8401701657498852871uint32(uint32_shiftr(Xb),code_integer_of_nat(N)),zero_zero_uint32) ).

% shiftr_uint32_code
tff(fact_9977_uint32__shiftl__def,axiom,
    ! [Xb: uint32,N: code_integer] :
      aa_Cod8401701657498852871uint32(uint32_shiftl(Xb),N) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,N),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),N) ),
        aa_Cod8401701657498852871uint32(aa_uin6552852722274082928uint32(aa_fun8838986389337097743uint32(undefi4150998591671624934uint32,bit_se5742574853984576102uint32),Xb),N),
        aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,aa_Code_integer_nat(code_nat_of_integer,N)),Xb) ) ).

% uint32_shiftl_def
tff(fact_9978_uint32__set__bit__def,axiom,
    ! [Xb: uint32,N: code_integer,B2: $o] :
      uint32_set_bit(Xb,N,(B2)) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,N),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),N) ),
        aa_o_uint32(aa_Cod7056071456392079164uint32(aa_uin937759467983514409uint32(aa_fun8001877339868165167uint32(undefi8935238026407452494uint32,generi1993664874377053279uint32),Xb),N),(B2)),
        aa_o_uint32(aa_nat_fun_o_uint32(aa_uin7041821913289200502uint32(generi1993664874377053279uint32,Xb),aa_Code_integer_nat(code_nat_of_integer,N)),(B2)) ) ).

% uint32_set_bit_def
tff(fact_9979_set__bit__uint32__code,axiom,
    ! [Xb: uint32,N: nat,B2: $o] :
      aa_o_uint32(aa_nat_fun_o_uint32(aa_uin7041821913289200502uint32(generi1993664874377053279uint32,Xb),N),(B2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),uint32_set_bit(Xb,code_integer_of_nat(N),(B2)),Xb) ).

% set_bit_uint32_code
tff(fact_9980_shiftl__uint32__code,axiom,
    ! [N: nat,Xb: uint32] :
      aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,N),Xb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),aa_Cod8401701657498852871uint32(uint32_shiftl(Xb),code_integer_of_nat(N)),zero_zero_uint32) ).

% shiftl_uint32_code
tff(fact_9981_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      aa_Code_integer_nat(code_nat_of_integer,K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_arz,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_9982_num__of__integer__code,axiom,
    ! [K: code_integer] :
      aa_Code_integer_num(code_num_of_integer,K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_asa,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_9983_int__of__integer__code,axiom,
    ! [K: code_integer] :
      aa_Code_integer_int(code_int_of_integer,K) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))),
        $ite(K = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_asb,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_9984_zero__integer_Orep__eq,axiom,
    aa_Code_integer_int(code_int_of_integer,zero_z3403309356797280102nteger) = zero_zero_int ).

% zero_integer.rep_eq
tff(fact_9985_int__of__integer__numeral,axiom,
    ! [K: num] : aa_Code_integer_int(code_int_of_integer,numera6620942414471956472nteger(K)) = aa_num_int(numeral_numeral_int,K) ).

% int_of_integer_numeral
tff(fact_9986_one__integer_Orep__eq,axiom,
    aa_Code_integer_int(code_int_of_integer,one_one_Code_integer) = one_one_int ).

% one_integer.rep_eq
tff(fact_9987_divide__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% divide_integer.rep_eq
tff(fact_9988_less__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),Xaa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% less_integer.rep_eq
tff(fact_9989_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L)) ) ).

% integer_less_iff
tff(fact_9990_int__of__integer__less__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),Y) ) ).

% int_of_integer_less_iff
tff(fact_9991_int__of__integer__pow,axiom,
    ! [Xb: code_integer,N: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),N)) = aa_nat_int(power_power_int(aa_Code_integer_int(code_int_of_integer,Xb)),N) ).

% int_of_integer_pow
tff(fact_9992_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L)) ) ).

% integer_less_eq_iff
tff(fact_9993_less__eq__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),Xaa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% less_eq_integer.rep_eq
tff(fact_9994_bin__last__integer_Orep__eq,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(bits_b8758750999018896077nteger,Xb)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Code_integer_int(code_int_of_integer,Xb)) ) ).

% bin_last_integer.rep_eq
tff(fact_9995_bin__rest__integer_Orep__eq,axiom,
    ! [Xb: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(bits_b2549910563261871055nteger,Xb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% bin_rest_integer.rep_eq
tff(fact_9996_Bit__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: $o] : aa_Code_integer_int(code_int_of_integer,bits_Bit_integer(Xb,(Xaa))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(Xaa))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Code_integer_int(code_int_of_integer,Xb))) ).

% Bit_integer.rep_eq
tff(fact_9997_uint32__shiftr__code,axiom,
    ! [W: uint32,N: code_integer] :
      rep_uint322(aa_Cod8401701657498852871uint32(uint32_shiftr(W),N)) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,N),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),N) ),
        rep_uint322(aa_Cod8401701657498852871uint32(aa_uin6552852722274082928uint32(aa_fun8838986389337097743uint32(undefi4150998591671624934uint32,bit_se3964402333458159761uint32),W),N)),
        bit_se5176125413884933531l_num1(aa_Code_integer_nat(code_nat_of_integer,N),rep_uint322(W)) ) ).

% uint32_shiftr_code
tff(fact_9998_uint32__shiftl__code,axiom,
    ! [W: uint32,N: code_integer] :
      rep_uint322(aa_Cod8401701657498852871uint32(uint32_shiftl(W),N)) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,N),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),N) ),
        rep_uint322(aa_Cod8401701657498852871uint32(aa_uin6552852722274082928uint32(aa_fun8838986389337097743uint32(undefi4150998591671624934uint32,bit_se5742574853984576102uint32),W),N)),
        bit_se837345729053750000l_num1(aa_Code_integer_nat(code_nat_of_integer,N),rep_uint322(W)) ) ).

% uint32_shiftl_code
tff(fact_9999_zero__uint32_Orep__eq,axiom,
    rep_uint322(zero_zero_uint32) = zero_z3563351764282998399l_num1 ).

% zero_uint32.rep_eq
tff(fact_10000_uint32_Oeven__iff__word__of,axiom,
    ! [P2: uint32] :
      ( aa_uint32_o(dvd_dvd_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),P2)
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),rep_uint322(P2)) ) ).

% uint32.even_iff_word_of
tff(fact_10001_size__uint32_Orep__eq,axiom,
    ! [Xb: uint32] : size_size_uint32(Xb) = size_s8261804613246490634l_num1(rep_uint322(Xb)) ).

% size_uint32.rep_eq
tff(fact_10002_uint32_Osize__eq__word__of,axiom,
    ! [P2: uint32] : size_size_uint32(P2) = size_s8261804613246490634l_num1(rep_uint322(P2)) ).

% uint32.size_eq_word_of
tff(fact_10003_uint32_Oless__iff__word__of,axiom,
    ! [P2: uint32,Q2: uint32] :
      ( aa_uint32_o(ord_less_uint32(P2),Q2)
    <=> aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(rep_uint322(P2)),rep_uint322(Q2)) ) ).

% uint32.less_iff_word_of
tff(fact_10004_less__uint32_Orep__eq,axiom,
    ! [Xb: uint32,Xaa: uint32] :
      ( aa_uint32_o(ord_less_uint32(Xb),Xaa)
    <=> aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(rep_uint322(Xb)),rep_uint322(Xaa)) ) ).

% less_uint32.rep_eq
tff(fact_10005_uint32__test__bit__code,axiom,
    ! [W: uint32,N: code_integer] :
      ( aa_Code_integer_o(uint32_test_bit(W),N)
    <=> $ite(
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,N),zero_z3403309356797280102nteger)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),N) ),
          aa_Code_integer_o(aa_uin5206268876086501546eger_o(aa_fun5262225984791465967eger_o(undefi9030187962712549518eger_o,bit_se5367290876889521763uint32),W),N),
          aa_nat_o(bit_se6859397288646540909l_num1(rep_uint322(W)),aa_Code_integer_nat(code_nat_of_integer,N)) ) ) ).

% uint32_test_bit_code
tff(fact_10006_uint32__set__bit__code,axiom,
    ! [W: uint32,N: code_integer,B2: $o] :
      rep_uint322(uint32_set_bit(W,N,(B2))) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,N),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),N) ),
        rep_uint322(aa_o_uint32(aa_Cod7056071456392079164uint32(aa_uin937759467983514409uint32(aa_fun8001877339868165167uint32(undefi8935238026407452494uint32,generi1993664874377053279uint32),W),N),(B2))),
        generi5268133209446125161l_num1(rep_uint322(W),aa_Code_integer_nat(code_nat_of_integer,N),(B2)) ) ).

% uint32_set_bit_code
tff(fact_10007_Uint32__signed__code,axiom,
    ! [I: code_integer] :
      rep_uint322(aa_Cod8401701657498852871uint32(uint32_signed,I)) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))))))))))))))))))))
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))))))))))))))))))),I) ),
        rep_uint322(aa_Cod8401701657498852871uint32(aa_fun8111848626357839434uint32(undefi1969809534184926113uint32,uint322),I)),
        ring_17408606157368542149l_num1(aa_Code_integer_int(code_I935103866777955880mbolic,I)) ) ).

% Uint32_signed_code
tff(fact_10008_uint32_Oset__bits__aux__code,axiom,
    ! [Fa: fun_nat_o,N: nat,W: uint32] :
      set_bits_aux_uint32(Fa,N,W) = $ite(
        N = zero_zero_nat,
        W,
        $let(
          n2: nat,
          n2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),
          set_bits_aux_uint32(Fa,n2,
            bit_se2966626333419230250uint32(aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,one_one_nat),W),
              $ite(aa_nat_o(Fa,n2),one_one_uint32,zero_zero_uint32))) ) ) ).

% uint32.set_bits_aux_code
tff(fact_10009_zero__uint32_Orsp,axiom,
    zero_z3563351764282998399l_num1 = zero_z3563351764282998399l_num1 ).

% zero_uint32.rsp
tff(fact_10010_uint32_Oset__bits__code,axiom,
    ! [P: fun_nat_o] : bit_bi705532357378895591uint32(P) = set_bits_aux_uint32(P,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))),zero_zero_uint32) ).

% uint32.set_bits_code
tff(fact_10011_int__of__integer__symbolic__aux__code_I1_J,axiom,
    aa_Code_integer_int(code_I935103866777955880mbolic,zero_z3403309356797280102nteger) = zero_zero_int ).

% int_of_integer_symbolic_aux_code(1)
tff(fact_10012_uint32__divmod__code,axiom,
    ! [Xb: uint32,Y: uint32] :
      uint32_divmod(Xb,Y) = $ite(
        aa_uint32_o(ord_less_eq_uint32(numera9087168376688890119uint32(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))))))))))))))))))),Y),
        $ite(aa_uint32_o(ord_less_uint32(Xb),Y),produc1400373151660368625uint32(zero_zero_uint32,Xb),produc1400373151660368625uint32(one_one_uint32,minus_minus_uint32(Xb,Y))),
        $ite(
          Y = zero_zero_uint32,
          produc1400373151660368625uint32(div0_uint32(Xb),mod0_uint32(Xb)),
          $let(
            q: uint32,
            q:= aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se5742574853984576102uint32,one_one_nat),uint32_sdiv(aa_uint32_uint32(aa_nat4578243230388868707uint32(bit_se3964402333458159761uint32,one_one_nat),Xb),Y)),
            $let(
              r: uint32,
              r:= minus_minus_uint32(Xb,aa_uint32_uint32(times_times_uint32(q),Y)),
              $ite(aa_uint32_o(ord_less_eq_uint32(Y),r),produc1400373151660368625uint32(plus_plus_uint32(q,one_one_uint32),minus_minus_uint32(r,Y)),produc1400373151660368625uint32(q,r)) ) ) ) ) ).

% uint32_divmod_code
tff(fact_10013_uint32__divmod__def,axiom,
    ! [Xb: uint32,Y: uint32] :
      uint32_divmod(Xb,Y) = $ite(Y = zero_zero_uint32,produc1400373151660368625uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(aa_fun8139942314387009082uint32(undefi5883245521155754897uint32,divide_divide_uint32),Xb),zero_zero_uint32),aa_uint32_uint32(aa_uin3778660748068847871uint32(aa_fun8139942314387009082uint32(undefi5883245521155754897uint32,modulo_modulo_uint32),Xb),zero_zero_uint32)),produc1400373151660368625uint32(aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,Xb),Y),aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,Xb),Y))) ).

% uint32_divmod_def
tff(fact_10014_div0__uint32__def,axiom,
    ! [Xb: uint32] : div0_uint32(Xb) = aa_uint32_uint32(aa_uin3778660748068847871uint32(aa_fun8139942314387009082uint32(undefi5883245521155754897uint32,divide_divide_uint32),Xb),zero_zero_uint32) ).

% div0_uint32_def
tff(fact_10015_mod0__uint32__def,axiom,
    ! [Xb: uint32] : mod0_uint32(Xb) = aa_uint32_uint32(aa_uin3778660748068847871uint32(aa_fun8139942314387009082uint32(undefi5883245521155754897uint32,modulo_modulo_uint32),Xb),zero_zero_uint32) ).

% mod0_uint32_def
tff(fact_10016_uint32__sdiv__code,axiom,
    ! [Xb: uint32,Y: uint32] :
      rep_uint322(uint32_sdiv(Xb,Y)) = $ite(Y = zero_zero_uint32,rep_uint322(aa_uint32_uint32(aa_uin3778660748068847871uint32(aa_fun8139942314387009082uint32(undefi5883245521155754897uint32,divide_divide_uint32),Xb),zero_zero_uint32)),signed6753297604338940182l_num1(rep_uint322(Xb),rep_uint322(Y))) ).

% uint32_sdiv_code
tff(fact_10017_sshiftr__uint32__code,axiom,
    ! [N: nat,Xb: uint32] :
      aa_uint32_uint32(aa_nat4578243230388868707uint32(signed489701013188660438uint32,N),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),
        aa_Cod8401701657498852871uint32(uint32_sshiftr(Xb),code_integer_of_nat(N)),
        $ite(aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),uminus_uminus_uint32(one_one_uint32),zero_zero_uint32) ) ).

% sshiftr_uint32_code
tff(fact_10018_integer__shiftl__def,axiom,
    ! [Xb: code_integer,N: code_integer] :
      aa_Cod1783974652533396728nteger(bits_integer_shiftl(Xb),N) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,N),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(undefi3911151285678701225nteger,Xb),N),aa_Cod1783974652533396728nteger(bit_se7788150548672797655nteger(aa_Code_integer_nat(code_nat_of_integer,N)),Xb)) ).

% integer_shiftl_def
tff(fact_10019_integer__shiftl__code_I2_J,axiom,
    ! [Xb: code_integer] : aa_Cod1783974652533396728nteger(bits_integer_shiftl(Xb),zero_z3403309356797280102nteger) = Xb ).

% integer_shiftl_code(2)
tff(fact_10020_uint32__sshiftr__def,axiom,
    ! [Xb: uint32,N: code_integer] :
      aa_Cod8401701657498852871uint32(uint32_sshiftr(Xb),N) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,N),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),N) ),
        aa_uint32_uint32(aa_Cod3565511682404152816uint32(aa_fun7349710212561521167uint32(undefi424500106705823974uint32,signed489701013188660438uint32),N),Xb),
        aa_uint32_uint32(aa_nat4578243230388868707uint32(signed489701013188660438uint32,aa_Code_integer_nat(code_nat_of_integer,N)),Xb) ) ).

% uint32_sshiftr_def
tff(fact_10021_uint32__sshiftr__code,axiom,
    ! [W: uint32,N: code_integer] :
      rep_uint322(aa_Cod8401701657498852871uint32(uint32_sshiftr(W),N)) = $ite(
        ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,N),zero_z3403309356797280102nteger)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))),N) ),
        rep_uint322(aa_uint32_uint32(aa_Cod3565511682404152816uint32(aa_fun7349710212561521167uint32(undefi424500106705823974uint32,signed489701013188660438uint32),N),W)),
        signed5000768011106662067l_num1(aa_Code_integer_nat(code_nat_of_integer,N),rep_uint322(W)) ) ).

% uint32_sshiftr_code
tff(fact_10022_div__uint32__code,axiom,
    ! [Xb: uint32,Y: uint32] :
      aa_uint32_uint32(aa_uin3778660748068847871uint32(divide_divide_uint32,Xb),Y) = $ite(Y = zero_zero_uint32,zero_zero_uint32,uint32_div(Xb,Y)) ).

% div_uint32_code
tff(fact_10023_mod__uint32__code,axiom,
    ! [Xb: uint32,Y: uint32] :
      aa_uint32_uint32(aa_uin3778660748068847871uint32(modulo_modulo_uint32,Xb),Y) = $ite(Y = zero_zero_uint32,Xb,uint32_mod(Xb,Y)) ).

% mod_uint32_code
tff(fact_10024_integer__shiftr__def,axiom,
    ! [Xb: code_integer,N: code_integer] :
      aa_Cod1783974652533396728nteger(bits_integer_shiftr(Xb),N) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,N),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(undefi3911151285678701225nteger,Xb),N),aa_Cod1783974652533396728nteger(bit_se3928097537394005634nteger(aa_Code_integer_nat(code_nat_of_integer,N)),Xb)) ).

% integer_shiftr_def
tff(fact_10025_integer__shiftr__code_I2_J,axiom,
    ! [Xb: code_integer] : aa_Cod1783974652533396728nteger(bits_integer_shiftr(Xb),zero_z3403309356797280102nteger) = Xb ).

% integer_shiftr_code(2)
tff(fact_10026_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
          aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),
            $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_asc(L),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_10027_integer__of__uint32__code,axiom,
    ! [N: uint32] :
      aa_uin2106300854182906631nteger(integer_of_uint32,N) = $ite(aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_Cod1783974652533396728nteger(bit_se1080825931792720795nteger(intege5370686899274169573signed(bit_se6294004230839889034uint32(N,numera9087168376688890119uint32(aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))))))))))))))))))))))))))))))),numera6620942414471956472nteger(aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))))))))))))))))))))))))),intege5370686899274169573signed(N)) ).

% integer_of_uint32_code
tff(fact_10028_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : code_divmod_abs(zero_z3403309356797280102nteger,J) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ).

% divmod_abs_code(6)
tff(fact_10029_integer__of__uint32__signed__def,axiom,
    ! [N: uint32] :
      intege5370686899274169573signed(N) = $ite(aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_uin2106300854182906631nteger(aa_fun3533256999457416010nteger(undefi4551866269130008737nteger,integer_of_uint32),N),aa_uin2106300854182906631nteger(integer_of_uint32,N)) ).

% integer_of_uint32_signed_def
tff(fact_10030_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : code_divmod_abs(J,zero_z3403309356797280102nteger) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,J)) ).

% divmod_abs_code(5)
tff(fact_10031_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L),
          $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_asd(L),code_divmod_abs(K,L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K),
            aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
              $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_ase(L),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_10032_integer__of__uint32__signed__code,axiom,
    ! [N: uint32] :
      intege5370686899274169573signed(N) = $ite(aa_nat_o(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_uin2106300854182906631nteger(aa_fun3533256999457416010nteger(undefi4551866269130008737nteger,integer_of_uint32),N),aa_int_Code_integer(code_integer_of_int,semiri7338730514057886004m1_int(rep_uint32(N)))) ).

% integer_of_uint32_signed_code
tff(fact_10033_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_z3403309356797280102nteger,produc6677183202524767010eger_o(zero_z3403309356797280102nteger,$false),produc9125791028180074456eger_o(aTP_Lamm_asf(K),code_divmod_abs(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_10034_Rep__uint32_H__code,axiom,
    ! [Xb: uint32] : rep_uint32(Xb) = bit_bi5746210779246519537l_num1(aa_uint32_fun_nat_o(bit_se5367290876889521763uint32,Xb)) ).

% Rep_uint32'_code
tff(fact_10035_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = produc6677183202524767010eger_o(aa_Cod1783974652533396728nteger(divide6298287555418463151nteger(K),numera6620942414471956472nteger(aa_num_num(bit0,one))),~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),K)) ).

% bit_cut_integer_def
tff(fact_10036_char__of__integer__code,axiom,
    ! [K: code_integer] : char_of_integer(K) = produc4188289175737317920o_char(aTP_Lamm_asn,code_bit_cut_integer(K)) ).

% char_of_integer_code
tff(fact_10037_uint32__sdiv__def,axiom,
    ! [Xb: uint32,Y: uint32] :
      uint32_sdiv(Xb,Y) = $ite(Y = zero_zero_uint32,aa_uint32_uint32(aa_uin3778660748068847871uint32(aa_fun8139942314387009082uint32(undefi5883245521155754897uint32,divide_divide_uint32),Xb),zero_zero_uint32),abs_uint32(signed6753297604338940182l_num1(rep_uint322(Xb),rep_uint322(Y)))) ).

% uint32_sdiv_def
tff(fact_10038_less__uint32_Oabs__eq,axiom,
    ! [Xaa: word_N3645301735248828278l_num1,Xb: word_N3645301735248828278l_num1] :
      ( aa_uint32_o(ord_less_uint32(abs_uint32(Xaa)),abs_uint32(Xb))
    <=> aa_wor4452366487969921764num1_o(ord_le750835935415966154l_num1(Xaa),Xb) ) ).

% less_uint32.abs_eq
tff(fact_10039_size__uint32_Oabs__eq,axiom,
    ! [Xb: word_N3645301735248828278l_num1] : size_size_uint32(abs_uint32(Xb)) = size_s8261804613246490634l_num1(Xb) ).

% size_uint32.abs_eq
tff(fact_10040_zero__uint32__def,axiom,
    zero_zero_uint32 = abs_uint32(zero_z3563351764282998399l_num1) ).

% zero_uint32_def
tff(fact_10041_DERIV__real__root__generic,axiom,
    ! [N: nat,Xb: real,D4: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( ( Xb != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
             => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
               => ( D4 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
               => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),D4,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_10042_DERIV__even__real__root,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_10043_inj__on__diff__nat,axiom,
    ! [N2: set_nat,K: nat] :
      ( ! [N3: nat] :
          ( aa_set_nat_o(member_nat(N3),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N3) )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_arn,K)),N2) ) ).

% inj_on_diff_nat
tff(fact_10044_inj__Suc,axiom,
    ! [N2: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N2) ).

% inj_Suc
tff(fact_10045_DERIV__mirror,axiom,
    ! [Fa: fun_real_real,Y: real,Xb: real] :
      ( has_fi5821293074295781190e_real(Fa,Y,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,Xb),top_top_set_real))
    <=> has_fi5821293074295781190e_real(aTP_Lamm_aso(Fa),aa_real_real(uminus_uminus_real,Y),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_mirror
tff(fact_10046_DERIV__ln__divide,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(divide_divide_real(one_one_real),Xb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_10047_DERIV__const__average,axiom,
    ! [A2: real,B2: real,V: fun_real_real,K: real] :
      ( ( A2 != B2 )
     => ( ! [X2: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X2,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),numeral_numeral_real(aa_num_num(bit0,one)))) = aa_real_real(divide_divide_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(V,A2)),aa_real_real(V,B2))),numeral_numeral_real(aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_10048_DERIV__ln,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_10049_DERIV__neg__dec__right,axiom,
    ! [Fa: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(Fa,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(Fa,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))),aa_real_real(Fa,Xb)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_10050_DERIV__pos__inc__right,axiom,
    ! [Fa: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(Fa,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(Fa,Xb)),aa_real_real(Fa,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_10051_DERIV__isconst__all,axiom,
    ! [Fa: fun_real_real,Xb: real,Y: real] :
      ( ! [X2: real] : has_fi5821293074295781190e_real(Fa,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real))
     => ( aa_real_real(Fa,Xb) = aa_real_real(Fa,Y) ) ) ).

% DERIV_isconst_all
tff(fact_10052_has__real__derivative__neg__dec__right,axiom,
    ! [Fa: fun_real_real,L: real,Xb: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(Fa,L,topolo2177554685111907308n_real(Xb,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3)),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(Fa,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))),aa_real_real(Fa,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_10053_has__real__derivative__pos__inc__right,axiom,
    ! [Fa: fun_real_real,L: real,Xb: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(Fa,L,topolo2177554685111907308n_real(Xb,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3)),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(Fa,Xb)),aa_real_real(Fa,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_10054_DERIV__pow,axiom,
    ! [N: nat,Xb: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_asp,N),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(Xb,S)) ).

% DERIV_pow
tff(fact_10055_has__real__derivative__pos__inc__left,axiom,
    ! [Fa: fun_real_real,L: real,Xb: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(Fa,L,topolo2177554685111907308n_real(Xb,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_set_real_o(member_real(aa_real_real(minus_minus_real(Xb),H3)),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(Fa,aa_real_real(minus_minus_real(Xb),H3))),aa_real_real(Fa,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_10056_has__real__derivative__neg__dec__left,axiom,
    ! [Fa: fun_real_real,L: real,Xb: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(Fa,L,topolo2177554685111907308n_real(Xb,S3))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_set_real_o(member_real(aa_real_real(minus_minus_real(Xb),H3)),S3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(Fa,Xb)),aa_real_real(Fa,aa_real_real(minus_minus_real(Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_10057_deriv__nonneg__imp__mono,axiom,
    ! [A2: real,B2: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_or1222579329274155063t_real(A2,B2))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) )
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),set_or1222579329274155063t_real(A2,B2))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X2)) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(G,A2)),aa_real_real(G,B2)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_10058_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A2: real,B2: real,Fa: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(Fa,Y4,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y4) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(Fa,A2)),aa_real_real(Fa,B2)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_10059_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A2: real,B2: real,Fa: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(Fa,Y4,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(Fa,B2)),aa_real_real(Fa,A2)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_10060_DERIV__local__const,axiom,
    ! [Fa: fun_real_real,L: real,Xb: real,D: real] :
      ( has_fi5821293074295781190e_real(Fa,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),Y2))),D)
             => ( aa_real_real(Fa,Xb) = aa_real_real(Fa,Y2) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_10061_DERIV__pos__inc__left,axiom,
    ! [Fa: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(Fa,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(Fa,aa_real_real(minus_minus_real(Xb),H3))),aa_real_real(Fa,Xb)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_10062_DERIV__neg__dec__left,axiom,
    ! [Fa: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(Fa,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(Fa,Xb)),aa_real_real(Fa,aa_real_real(minus_minus_real(Xb),H3))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_10063_DERIV__local__min,axiom,
    ! [Fa: fun_real_real,L: real,Xb: real,D: real] :
      ( has_fi5821293074295781190e_real(Fa,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),Y2))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(Fa,Xb)),aa_real_real(Fa,Y2)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_10064_DERIV__local__max,axiom,
    ! [Fa: fun_real_real,L: real,Xb: real,D: real] :
      ( has_fi5821293074295781190e_real(Fa,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),Y2))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(Fa,Y2)),aa_real_real(Fa,Xb)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_10065_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B2: real,Fa: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(Fa,Y4,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(Fa,A2)),aa_real_real(Fa,B2)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_10066_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B2: real,Fa: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(Fa,Y4,topolo2177554685111907308n_real(X2,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(Fa,B2)),aa_real_real(Fa,A2)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_10067_MVT2,axiom,
    ! [A2: real,B2: real,Fa: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B2)
             => has_fi5821293074295781190e_real(Fa,aa_real_real(F,X2),topolo2177554685111907308n_real(X2,top_top_set_real)) ) )
       => ? [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Z2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B2)
            & ( aa_real_real(minus_minus_real(aa_real_real(Fa,B2)),aa_real_real(Fa,A2)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(B2),A2)),aa_real_real(F,Z2)) ) ) ) ) ).

% MVT2
tff(fact_10068_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,M: real,Xb: real,N: nat] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_asq(G),N),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)))),M),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_10069_has__real__derivative__powr,axiom,
    ! [Z: real,R: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => has_fi5821293074295781190e_real(aTP_Lamm_asr(R),aa_real_real(times_times_real(R),aa_real_real(powr_real(Z),aa_real_real(minus_minus_real(R),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_10070_summable__reindex,axiom,
    ! [Fa: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(Fa)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Fa,X2))
         => summable_real(comp_nat_real_nat(Fa,G)) ) ) ) ).

% summable_reindex
tff(fact_10071_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,M: real,Xb: real,R: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xb))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_ass(G),R),aa_real_real(times_times_real(aa_real_real(times_times_real(R),aa_real_real(powr_real(aa_real_real(G,Xb)),aa_real_real(minus_minus_real(R),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_10072_DERIV__log,axiom,
    ! [Xb: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(log(B2),aa_real_real(divide_divide_real(one_one_real),aa_real_real(times_times_real(aa_real_real(ln_ln_real,B2)),Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_log
tff(fact_10073_DERIV__powr,axiom,
    ! [G: fun_real_real,M: real,Xb: real,Fa: fun_real_real,R: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xb))
       => ( has_fi5821293074295781190e_real(Fa,R,topolo2177554685111907308n_real(Xb,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_ast(G),Fa),aa_real_real(times_times_real(aa_real_real(powr_real(aa_real_real(G,Xb)),aa_real_real(Fa,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(times_times_real(R),aa_real_real(ln_ln_real,aa_real_real(G,Xb)))),aa_real_real(divide_divide_real(aa_real_real(times_times_real(M),aa_real_real(Fa,Xb))),aa_real_real(G,Xb)))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_10074_DERIV__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),numeral_numeral_real(aa_num_num(bit0,one))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_10075_DERIV__arctan,axiom,
    ! [Xb: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ).

% DERIV_arctan
tff(fact_10076_arsinh__real__has__field__derivative,axiom,
    ! [Xb: real,A3: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xb,A3)) ).

% arsinh_real_has_field_derivative
tff(fact_10077_DERIV__real__sqrt__generic,axiom,
    ! [Xb: real,D4: real] :
      ( ( Xb != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( D4 = aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),numeral_numeral_real(aa_num_num(bit0,one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
           => ( D4 = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb)))),numeral_numeral_real(aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D4,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_10078_arcosh__real__has__field__derivative,axiom,
    ! [Xb: real,A3: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xb,A3)) ) ).

% arcosh_real_has_field_derivative
tff(fact_10079_artanh__real__has__field__derivative,axiom,
    ! [Xb: real,A3: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xb,A3)) ) ).

% artanh_real_has_field_derivative
tff(fact_10080_suminf__reindex__mono,axiom,
    ! [Fa: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(Fa)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Fa,X2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(comp_nat_real_nat(Fa,G))),suminf_real(Fa)) ) ) ) ).

% suminf_reindex_mono
tff(fact_10081_DERIV__real__root,axiom,
    ! [N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_10082_DERIV__arccos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => has_fi5821293074295781190e_real(arccos,aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_10083_DERIV__arcsin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_10084_inj__on__char__of__nat,axiom,
    aa_set_nat_o(inj_on_nat_char(unique3096191561947761185of_nat),aa_nat_set_nat(aa_nat1818925681583257441et_nat(set_or4665077453230672383an_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% inj_on_char_of_nat
tff(fact_10085_suminf__reindex,axiom,
    ! [Fa: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(Fa)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Fa,X2))
         => ( ! [X2: nat] :
                ( ~ aa_set_nat_o(member_nat(X2),image_nat_nat(G,top_top_set_nat))
               => ( aa_nat_real(Fa,X2) = zero_zero_real ) )
           => ( suminf_real(comp_nat_real_nat(Fa,G)) = suminf_real(Fa) ) ) ) ) ) ).

% suminf_reindex
tff(fact_10086_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,Fa: fun_real_real,Xb: real,N: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = Fa )
        & ! [M3: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X2),topolo2177554685111907308n_real(X2,top_top_set_real)) )
     => ? [T: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb))
          & ( aa_real_real(Fa,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_asu(Diff),Xb),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_10087_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,Fa: fun_real_real,Xb: real,N: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = Fa )
     => ( ! [M3: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X2),topolo2177554685111907308n_real(X2,top_top_set_real))
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(Fa,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_asu(Diff),Xb),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_10088_DERIV__odd__real__root,axiom,
    ! [N: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
     => ( ( Xb != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,N),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_10089_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,Fa: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),zero_zero_real)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = Fa )
         => ( ! [M3: nat,T: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
           => ? [T: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T),zero_zero_real)
                & ( aa_real_real(Fa,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_asv(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_10090_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,Fa: fun_real_real,N: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = Fa )
       => ( ! [M3: nat,T: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),H) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
         => ? [T: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),H)
              & ( aa_real_real(Fa,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_asv(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_10091_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: fun_na6724076748692969250l_real,Fa: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = Fa )
         => ( ! [M3: nat,T: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),H) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
           => ? [T: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T),H)
                & ( aa_real_real(Fa,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_asv(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_10092_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,Fa: fun_real_real,N: nat,Xb: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = Fa )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( ( Xb != zero_zero_real )
         => ( ! [M3: nat,X2: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X2),topolo2177554685111907308n_real(X2,top_top_set_real))
           => ? [T: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb))
                & ( aa_real_real(Fa,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_asu(Diff),Xb),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_10093_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,Fa: fun_real_real,N: nat,Xb: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = Fa )
     => ( ! [M3: nat,T: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(Fa,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_asu(Diff),Xb),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_10094_Taylor,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,Fa: fun_real_real,A2: real,B2: real,C: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = Fa )
       => ( ! [M3: nat,T: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),B2) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),Xb)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),B2)
                 => ( ( Xb != C )
                   => ? [T: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),C),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),C) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),Xb) ) )
                        & ( aa_real_real(Fa,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_asw(Diff),C),Xb),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Xb),C)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_10095_Taylor__up,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,Fa: fun_real_real,A2: real,B2: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = Fa )
       => ( ! [M3: nat,T: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),B2) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B2)
             => ? [T: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T),B2)
                  & ( aa_real_real(Fa,B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_asx(Diff),B2),C),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(B2),C)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_10096_Taylor__down,axiom,
    ! [N: nat,Diff: fun_na6724076748692969250l_real,Fa: fun_real_real,A2: real,B2: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = Fa )
       => ( ! [M3: nat,T: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),B2) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B2)
             => ? [T: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),T)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T),C)
                  & ( aa_real_real(Fa,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_asx(Diff),A2),C),aa_nat_set_nat(set_ord_lessThan_nat,N))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,N),T)),aa_nat_real(semiri2265585572941072030t_real,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(A2),C)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_10097_inj__sgn__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => aa_set_real_o(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_aqq,N)),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_10098_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B5: real] :
      ( ! [M3: nat,T: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),H) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
     => ( ( N = aa_nat_nat(suc,K) )
       => ! [M2: nat,T6: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_asz(N),Diff),B5),M2),aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T6)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ata(Diff),M2),T6),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2))))),aa_real_real(times_times_real(B5),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,T6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T6,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_10099_DERIV__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_atb,suminf_real(aa_real_fun_nat_real(aTP_Lamm_atc,Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_10100_DERIV__power__series_H,axiom,
    ! [R4: real,Fa: fun_nat_real,X0: real] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R4),R4))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_atd(Fa),X2)) )
     => ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R4),R4))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
         => has_fi5821293074295781190e_real(aTP_Lamm_atf(Fa),suminf_real(aa_real_fun_nat_real(aTP_Lamm_atd(Fa),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_10101_DERIV__isconst3,axiom,
    ! [A2: real,B2: real,Xb: real,Y: real,Fa: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_set_real_o(member_real(Xb),set_or1633881224788618240n_real(A2,B2))
       => ( aa_set_real_o(member_real(Y),set_or1633881224788618240n_real(A2,B2))
         => ( ! [X2: real] :
                ( aa_set_real_o(member_real(X2),set_or1633881224788618240n_real(A2,B2))
               => has_fi5821293074295781190e_real(Fa,zero_zero_real,topolo2177554685111907308n_real(X2,top_top_set_real)) )
           => ( aa_real_real(Fa,Xb) = aa_real_real(Fa,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_10102_DERIV__series_H,axiom,
    ! [Fa: fun_re4238438383274954274t_real,F: fun_re4238438383274954274t_real,X0: real,A2: real,B2: real,L5: fun_nat_real] :
      ( ! [N3: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_alc(Fa),N3),aa_nat_real(aa_real_fun_nat_real(F,X0),N3),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),set_or1633881224788618240n_real(A2,B2))
           => summable_real(aa_real_fun_nat_real(Fa,X2)) )
       => ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(A2,B2))
         => ( summable_real(aa_real_fun_nat_real(F,X0))
           => ( summable_real(L5)
             => ( ! [N3: nat,X2: real,Y2: real] :
                    ( aa_set_real_o(member_real(X2),set_or1633881224788618240n_real(A2,B2))
                   => ( aa_set_real_o(member_real(Y2),set_or1633881224788618240n_real(A2,B2))
                     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(Fa,X2),N3)),aa_nat_real(aa_real_fun_nat_real(Fa,Y2),N3)))),aa_real_real(times_times_real(aa_nat_real(L5,N3)),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X2),Y2)))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_amk(Fa),suminf_real(aa_real_fun_nat_real(F,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_10103_LIM__fun__less__zero,axiom,
    ! [Fa: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(Fa,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(C),X3))),R3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(Fa,X3)),zero_zero_real) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_10104_LIM__fun__not__zero,axiom,
    ! [Fa: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(Fa,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(C),X3))),R3) )
               => ( aa_real_real(Fa,X3) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_10105_LIM__fun__gt__zero,axiom,
    ! [Fa: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(Fa,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(C),X3))),R3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(Fa,X3)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_10106_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or5832277885323065728an_int(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_10107_isCont__inverse__function2,axiom,
    ! [A2: real,Xb: real,B2: real,G: fun_real_real,Fa: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B2)
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),Z2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B2)
               => ( aa_real_real(G,aa_real_real(Fa,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),Z2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B2)
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),Fa) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(Fa,Xb),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_10108_isCont__ln,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero_real )
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),ln_ln_real) ) ).

% isCont_ln
tff(fact_10109_isCont__arcosh,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_10110_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_atg,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_10111_DERIV__inverse__function,axiom,
    ! [Fa: fun_real_real,D4: real,G: fun_real_real,Xb: real,A2: real,B2: real] :
      ( has_fi5821293074295781190e_real(Fa,D4,topolo2177554685111907308n_real(aa_real_real(G,Xb),top_top_set_real))
     => ( ( D4 != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B2)
           => ( ! [Y2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Y2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),B2)
                   => ( aa_real_real(Fa,aa_real_real(G,Y2)) = Y2 ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),G)
               => has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D4),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_10112_isCont__arccos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_10113_isCont__arcsin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_10114_LIM__less__bound,axiom,
    ! [B2: real,Xb: real,Fa: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),Xb)
     => ( ! [X2: real] :
            ( aa_set_real_o(member_real(X2),set_or1633881224788618240n_real(B2,Xb))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Fa,X2)) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),Fa)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Fa,Xb)) ) ) ) ).

% LIM_less_bound
tff(fact_10115_isCont__artanh,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_10116_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5832277885323065728an_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_10117_isCont__inverse__function,axiom,
    ! [D: real,Xb: real,G: fun_real_real,Fa: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
     => ( ! [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z2),Xb))),D)
           => ( aa_real_real(G,aa_real_real(Fa,Z2)) = Z2 ) )
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z2),Xb))),D)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),Fa) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(Fa,Xb),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_10118_GMVT_H,axiom,
    ! [A2: real,B2: real,Fa: fun_real_real,G: fun_real_real,G2: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ! [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),Z2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B2)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),Fa) ) )
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),Z2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B2)
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),G) ) )
         => ( ! [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Z2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B2)
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
           => ( ! [Z2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Z2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B2)
                   => has_fi5821293074295781190e_real(Fa,aa_real_real(F,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
             => ? [C2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),C2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B2)
                  & ( aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(Fa,B2)),aa_real_real(Fa,A2))),aa_real_real(G2,C2)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(G,B2)),aa_real_real(G,A2))),aa_real_real(F,C2)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_10119_summable__Leibniz_I3_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(A2,zero_zero_nat)),zero_zero_real)
         => ! [N4: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_ath(A2))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_ath(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_ath(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_10120_summable__Leibniz_I2_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(A2,zero_zero_nat))
         => ! [N4: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_ath(A2))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_ath(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))),groups6591440286371151544t_real(aTP_Lamm_ath(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_10121_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_10122_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ji,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_10123_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_ati,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_10124_nested__sequence__unique,axiom,
    ! [Fa: fun_nat_real,G: fun_nat_real] :
      ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Fa,N3)),aa_nat_real(Fa,aa_nat_nat(suc,N3)))
     => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N3))),aa_nat_real(G,N3))
       => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Fa,N3)),aa_nat_real(G,N3))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vk(Fa),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L2: real] :
                ( ! [N4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Fa,N4)),L2)
                & filterlim_nat_real(Fa,topolo2815343760600316023s_real(L2),at_top_nat)
                & ! [N4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),aa_nat_real(G,N4))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_10125_LIMSEQ__inverse__zero,axiom,
    ! [X5: fun_nat_real] :
      ( ! [R3: real] :
        ? [N9: nat] :
        ! [N3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N9),N3)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R3),aa_nat_real(X5,N3)) )
     => filterlim_nat_real(aTP_Lamm_atj(X5),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_10126_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_atk,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_10127_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_atl,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_10128_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_atm,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_10129_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_atn,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_10130_increasing__LIMSEQ,axiom,
    ! [Fa: fun_nat_real,L: real] :
      ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Fa,N3)),aa_nat_real(Fa,aa_nat_nat(suc,N3)))
     => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Fa,N3)),L)
       => ( ! [E: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
             => ? [N4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Fa,N4)),E)) )
         => filterlim_nat_real(Fa,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_10131_LIMSEQ__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_10132_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xb: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ato(Xb),A2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_10133_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_10134_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_10135_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_atp,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_10136_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_atq,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_10137_tendsto__exp__limit__sequentially,axiom,
    ! [Xb: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_atr,Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_10138_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ats,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_10139_summable__Leibniz_I1_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => summable_real(aTP_Lamm_ath(A2)) ) ) ).

% summable_Leibniz(1)
tff(fact_10140_summable,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N3))
       => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N3))),aa_nat_real(A2,N3))
         => summable_real(aTP_Lamm_ath(A2)) ) ) ) ).

% summable
tff(fact_10141_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_att(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K2: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_atu(Theta),K2),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_10142_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_atv(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K2: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_atu(Theta),K2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_10143_summable__Leibniz_I4_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => filterlim_nat_real(aTP_Lamm_atw(A2),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ath(A2))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_10144_zeroseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_uk,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_10145_summable__Leibniz_H_I2_J,axiom,
    ! [A2: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N3))
       => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N3))),aa_nat_real(A2,N3))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_ath(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)))),suminf_real(aTP_Lamm_ath(A2))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_10146_summable__Leibniz_H_I3_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N3))
       => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N3))),aa_nat_real(A2,N3))
         => filterlim_nat_real(aTP_Lamm_atw(A2),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ath(A2))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_10147_sums__alternating__upper__lower,axiom,
    ! [A2: fun_nat_real] :
      ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N3))),aa_nat_real(A2,N3))
     => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N3))
       => ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L2: real] :
              ( ! [N4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_ath(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)))),L2)
              & filterlim_nat_real(aTP_Lamm_atw(A2),topolo2815343760600316023s_real(L2),at_top_nat)
              & ! [N4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),groups6591440286371151544t_real(aTP_Lamm_ath(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_atx(A2),topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_10148_summable__Leibniz_I5_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => filterlim_nat_real(aTP_Lamm_atx(A2),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ath(A2))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_10149_summable__Leibniz_H_I5_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N3))
       => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N3))),aa_nat_real(A2,N3))
         => filterlim_nat_real(aTP_Lamm_atx(A2),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ath(A2))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_10150_summable__Leibniz_H_I4_J,axiom,
    ! [A2: fun_nat_real,N: nat] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N3))
       => ( ! [N3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N3))),aa_nat_real(A2,N3))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_ath(A2))),groups6591440286371151544t_real(aTP_Lamm_ath(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_10151_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_arx(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_10152_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_aty(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_10153_le__sequentially,axiom,
    ! [F4: filter_nat] :
      ( ord_le2510731241096832064er_nat(F4,at_top_nat)
    <=> ! [N12: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N12),F4) ) ).

% le_sequentially
tff(fact_10154_eventually__sequentially,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N12: nat] :
        ! [N8: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N12),N8)
         => aa_nat_o(P,N8) ) ) ).

% eventually_sequentially
tff(fact_10155_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X2)
         => aa_nat_o(P,X2) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_10156_eventually__False__sequentially,axiom,
    ~ eventually_nat(aTP_Lamm_fs,at_top_nat) ).

% eventually_False_sequentially
tff(fact_10157_sequentially__offset,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_aty(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_10158_lhopital__left__at__top__at__top,axiom,
    ! [Fa: fun_real_real,A2: real,G: fun_real_real,F: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(Fa,at_top_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_atz(Fa),F),topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_atz(G),G2),topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aua(F),G2),at_top_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aua(Fa),G),at_top_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_10159_lhopital__at__top__at__top,axiom,
    ! [Fa: fun_real_real,A2: real,G: fun_real_real,F: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(Fa,at_top_real,topolo2177554685111907308n_real(A2,top_top_set_real))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A2,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_atz(Fa),F),topolo2177554685111907308n_real(A2,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_atz(G),G2),topolo2177554685111907308n_real(A2,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aua(F),G2),at_top_real,topolo2177554685111907308n_real(A2,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aua(Fa),G),at_top_real,topolo2177554685111907308n_real(A2,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_10160_exp__at__top,axiom,
    filterlim_real_real(exp_real,at_top_real,at_top_real) ).

% exp_at_top
tff(fact_10161_sqrt__at__top,axiom,
    filterlim_real_real(sqrt,at_top_real,at_top_real) ).

% sqrt_at_top
tff(fact_10162_ln__at__top,axiom,
    filterlim_real_real(ln_ln_real,at_top_real,at_top_real) ).

% ln_at_top
tff(fact_10163_eventually__at__left__real,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_aub(B2),A2),topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2))) ) ).

% eventually_at_left_real
tff(fact_10164_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,Xb: real,G2: fun_real_real,Fa: fun_real_real,F: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
     => ( eventually_real(aTP_Lamm_auc(G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_atz(Fa),F),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_atz(G),G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aud(G2),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aud(G),Fa),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_10165_lhospital__at__top__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,Fa: fun_real_real,F: fun_real_real,Xb: real] :
      ( filterlim_real_real(G,at_top_real,at_top_real)
     => ( eventually_real(aTP_Lamm_auc(G2),at_top_real)
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_atz(Fa),F),at_top_real)
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_atz(G),G2),at_top_real)
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aud(G2),F),topolo2815343760600316023s_real(Xb),at_top_real)
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aud(G),Fa),topolo2815343760600316023s_real(Xb),at_top_real) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_10166_lhopital__at__top,axiom,
    ! [G: fun_real_real,Xb: real,G2: fun_real_real,Fa: fun_real_real,F: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( eventually_real(aTP_Lamm_auc(G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_atz(Fa),F),topolo2177554685111907308n_real(Xb,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_atz(G),G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aud(G2),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xb,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aud(G),Fa),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_10167_filterlim__real__sequentially,axiom,
    filterlim_nat_real(semiri5074537144036343181t_real,at_top_real,at_top_nat) ).

% filterlim_real_sequentially
tff(fact_10168_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_aue,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_10169_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_auf,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_10170_lhopital,axiom,
    ! [Fa: fun_real_real,Xb: real,G: fun_real_real,G2: fun_real_real,F: fun_real_real,F4: filter_real] :
      ( filterlim_real_real(Fa,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,top_top_set_real))
       => ( eventually_real(aTP_Lamm_auc(G),topolo2177554685111907308n_real(Xb,top_top_set_real))
         => ( eventually_real(aTP_Lamm_auc(G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_atz(Fa),F),topolo2177554685111907308n_real(Xb,top_top_set_real))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_atz(G),G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aud(G2),F),F4,topolo2177554685111907308n_real(Xb,top_top_set_real))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aua(Fa),G),F4,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_10171_lhopital__left,axiom,
    ! [Fa: fun_real_real,Xb: real,G: fun_real_real,G2: fun_real_real,F: fun_real_real,F4: filter_real] :
      ( filterlim_real_real(Fa,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
       => ( eventually_real(aTP_Lamm_auc(G),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
         => ( eventually_real(aTP_Lamm_auc(G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_atz(Fa),F),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_atz(G),G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aud(G2),F),F4,topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aua(Fa),G),F4,topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_10172_tendsto__exp__limit__at__top,axiom,
    ! [Xb: real] : filterlim_real_real(aTP_Lamm_aug(Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_10173_filterlim__tan__at__left,axiom,
    filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))),set_or5984915006950818249n_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))))) ).

% filterlim_tan_at_left
tff(fact_10174_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_10175_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,Fa: fun_real_real,Flim: real] :
      ( ! [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),X2)
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(Fa,Y4,topolo2177554685111907308n_real(X2,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) )
     => ( filterlim_real_real(Fa,topolo2815343760600316023s_real(Flim),at_top_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(Fa,B2)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_10176_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ue,Uu) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_10177_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_aoe,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(one_one_real),numeral_numeral_real(aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_2
tff(fact_10178_ATP_Olambda__3,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_apy,Uu)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu),numeral_numeral_real(aa_num_num(bit0,one)))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_3
tff(fact_10179_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_aok,Uu) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uu)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu)) ).

% ATP.lambda_4
tff(fact_10180_ATP_Olambda__5,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_atg,Uu) = aa_real_real(divide_divide_real(aa_real_real(cos_real,Uu)),aa_real_real(sin_real,Uu)) ).

% ATP.lambda_5
tff(fact_10181_ATP_Olambda__6,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_aue,Uu) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Uu)),Uu) ).

% ATP.lambda_6
tff(fact_10182_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ati,Uu) = aa_real_real(aa_nat_fun_real_real(root,Uu),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_7
tff(fact_10183_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_arq,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_8
tff(fact_10184_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_atk,Uu) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_9
tff(fact_10185_ATP_Olambda__10,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_aqr,Uu)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uu) ) ).

% ATP.lambda_10
tff(fact_10186_ATP_Olambda__11,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_atb,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ul,Uu)) ).

% ATP.lambda_11
tff(fact_10187_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_atl,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_12
tff(fact_10188_ATP_Olambda__13,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_apl,Uu) = aa_num_option_num(some_num,aa_num_num(bit1,Uu)) ).

% ATP.lambda_13
tff(fact_10189_ATP_Olambda__14,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_apj,Uu) = aa_num_option_num(some_num,aa_num_num(bit0,Uu)) ).

% ATP.lambda_14
tff(fact_10190_ATP_Olambda__15,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_aqt,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_15
tff(fact_10191_ATP_Olambda__16,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_aqw,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_16
tff(fact_10192_ATP_Olambda__17,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_mo,Uu) = aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(suc,Uu)) ).

% ATP.lambda_17
tff(fact_10193_ATP_Olambda__18,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat] :
      ( aa_fun8910613404238449780_nat_o(aTP_Lamm_aor,Uu)
    <=> ! [X: vEBT_VEBT,Xa3: vEBT_VEBTi,Xb4: nat] : refine7594492741263601813on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_predi,Xa3),Xb4),aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(produc9121605208400662357on_nat(produc1757988346207259447on_nat(Uu)),X),Xa3),Xb4)) ) ).

% ATP.lambda_18
tff(fact_10194_ATP_Olambda__19,axiom,
    ! [Uu: nat] : aa_nat_option_num(aTP_Lamm_apn,Uu) = aa_num_option_num(some_num,one) ).

% ATP.lambda_19
tff(fact_10195_ATP_Olambda__20,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_apr(Uu),Uua) = aa_num_option_num(case_num_option_num(aa_num_option_num(some_num,one),aTP_Lamm_app(Uua),aTP_Lamm_apq(Uua)),Uu) ).

% ATP.lambda_20
tff(fact_10196_ATP_Olambda__21,axiom,
    ! [Uu: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_aof(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_21
tff(fact_10197_ATP_Olambda__22,axiom,
    ! [Uu: $o,Uua: $o] :
      aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(aTP_Lamm_aj,(Uu)),(Uua)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),xa),
        $ite(
          (Uua),
          heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),
          $ite((Uu),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)) ),
        $ite(
          xa = one_one_nat,
          $ite((Uu),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)),
          heap_T3487192422709364219on_nat(none_nat) ) ) ).

% ATP.lambda_22
tff(fact_10198_ATP_Olambda__23,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aqs,Uu),Uua) = $ite(Uu = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Uu)),Uua)),aa_int_int(abs_abs_int,Uu))) ).

% ATP.lambda_23
tff(fact_10199_ATP_Olambda__24,axiom,
    ! [Uu: option_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_fj(Uu),Uua)
    <=> case_option_o_int($true,aa_int_fun_int_o(aTP_Lamm_fi,Uua),Uu) ) ).

% ATP.lambda_24
tff(fact_10200_ATP_Olambda__25,axiom,
    ! [Uu: option_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_fh(Uu),Uua)
    <=> aa_option_nat_o(case_option_o_nat($true,aa_nat_fun_nat_o(aTP_Lamm_fg,Uua)),Uu) ) ).

% ATP.lambda_25
tff(fact_10201_ATP_Olambda__26,axiom,
    ! [Uu: option_num,Uua: num] :
      ( aa_num_o(aTP_Lamm_ff(Uu),Uua)
    <=> aa_option_num_o(case_option_o_num($true,aa_num_fun_num_o(aTP_Lamm_fe,Uua)),Uu) ) ).

% ATP.lambda_26
tff(fact_10202_ATP_Olambda__27,axiom,
    ! [Uu: option_rat,Uua: rat] :
      ( aa_rat_o(aTP_Lamm_fd(Uu),Uua)
    <=> case_option_o_rat($true,aa_rat_fun_rat_o(aTP_Lamm_fc,Uua),Uu) ) ).

% ATP.lambda_27
tff(fact_10203_ATP_Olambda__28,axiom,
    ! [Uu: option_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_fb(Uu),Uua)
    <=> case_option_o_real($true,aa_real_fun_real_o(aTP_Lamm_fa,Uua),Uu) ) ).

% ATP.lambda_28
tff(fact_10204_ATP_Olambda__29,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_aps,Uu),Uua) = aa_nat_option_num(case_nat_option_num(none_num,aTP_Lamm_apr(Uua)),Uu) ).

% ATP.lambda_29
tff(fact_10205_ATP_Olambda__30,axiom,
    ! [Uu: option_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ev(Uu),Uua)
    <=> case_o4401850862724306899et_nat($false,ord_less_eq_set_nat(Uua),Uu) ) ).

% ATP.lambda_30
tff(fact_10206_ATP_Olambda__31,axiom,
    ! [Uu: option_rat,Uua: rat] :
      ( aa_rat_o(aTP_Lamm_ew(Uu),Uua)
    <=> case_option_o_rat($false,aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu) ) ).

% ATP.lambda_31
tff(fact_10207_ATP_Olambda__32,axiom,
    ! [Uu: option_num,Uua: num] :
      ( aa_num_o(aTP_Lamm_ex(Uu),Uua)
    <=> aa_option_num_o(case_option_o_num($false,aa_num_fun_num_o(ord_less_eq_num,Uua)),Uu) ) ).

% ATP.lambda_32
tff(fact_10208_ATP_Olambda__33,axiom,
    ! [Uu: option_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ey(Uu),Uua)
    <=> aa_option_nat_o(case_option_o_nat($false,aa_nat_fun_nat_o(ord_less_eq_nat,Uua)),Uu) ) ).

% ATP.lambda_33
tff(fact_10209_ATP_Olambda__34,axiom,
    ! [Uu: option_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ez(Uu),Uua)
    <=> case_option_o_int($false,aa_int_fun_int_o(ord_less_eq_int,Uua),Uu) ) ).

% ATP.lambda_34
tff(fact_10210_ATP_Olambda__35,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_app(Uu),Uua) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_apj),aa_num_option_num(bit_take_bit_num(Uu),Uua)) ).

% ATP.lambda_35
tff(fact_10211_ATP_Olambda__36,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_apo(Uu),Uua) = aa_opt5369159329878751634on_num(case_o6005452278849405969um_num(none_num,aTP_Lamm_apj),aa_num_option_num(bit_take_bit_num(Uua),Uu)) ).

% ATP.lambda_36
tff(fact_10212_ATP_Olambda__37,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tt,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_37
tff(fact_10213_ATP_Olambda__38,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aol,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_38
tff(fact_10214_ATP_Olambda__39,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aoo,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_39
tff(fact_10215_ATP_Olambda__40,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_apz,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
        & ( aa_real_real(sin_real,Uua) = Uu ) ) ) ).

% ATP.lambda_40
tff(fact_10216_ATP_Olambda__41,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_apt,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),aa_real_real(divide_divide_real(pi),numeral_numeral_real(aa_num_num(bit0,one))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_41
tff(fact_10217_ATP_Olambda__42,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_asb,Uu),Uua) = $let(
        l2: int,
        l2:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_Code_integer_int(code_int_of_integer,Uu)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(aa_int_fun_int_int(plus_plus_int,l2),one_one_int)) ) ).

% ATP.lambda_42
tff(fact_10218_ATP_Olambda__43,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_mn,Uu),Uua) = $let(
        m2: code_integer,
        m2:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,Uu)),
        $ite(Uua = zero_zero_nat,m2,aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(m2),one_one_Code_integer)) ) ).

% ATP.lambda_43
tff(fact_10219_ATP_Olambda__44,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_uint32(aa_nat6935575117487981255uint32(aTP_Lamm_mh,Uu),Uua) = $let(
        m2: uint32,
        m2:= aa_uint32_uint32(times_times_uint32(numera9087168376688890119uint32(aa_num_num(bit0,one))),aa_nat_uint32(semiri2565882477558803405uint32,Uu)),
        $ite(Uua = zero_zero_nat,m2,plus_plus_uint32(m2,one_one_uint32)) ) ).

% ATP.lambda_44
tff(fact_10220_ATP_Olambda__45,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat5599461682457136004l_num1(aa_nat6251203789436639057l_num1(aTP_Lamm_mi,Uu),Uua) = $let(
        m2: word_N3645301735248828278l_num1,
        m2:= aa_wor7274019993978516266l_num1(times_7065122842183080059l_num1(numera7442385471795722001l_num1(aa_num_num(bit0,one))),aa_nat5599461682457136004l_num1(semiri8819519690708144855l_num1,Uu)),
        $ite(Uua = zero_zero_nat,m2,plus_p361126936061061375l_num1(m2,one_on7727431528512463931l_num1)) ) ).

% ATP.lambda_45
tff(fact_10221_ATP_Olambda__46,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mk,Uu),Uua) = $let(
        m2: real,
        m2:= aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Uu)),
        $ite(Uua = zero_zero_nat,m2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),one_one_real)) ) ).

% ATP.lambda_46
tff(fact_10222_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_mj,Uu),Uua) = $let(
        m2: rat,
        m2:= aa_rat_rat(times_times_rat(numeral_numeral_rat(aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu)),
        $ite(Uua = zero_zero_nat,m2,plus_plus_rat(m2,one_one_rat)) ) ).

% ATP.lambda_47
tff(fact_10223_ATP_Olambda__48,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mm,Uu),Uua) = $let(
        m2: nat,
        m2:= aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Uu)),
        $ite(Uua = zero_zero_nat,m2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),one_one_nat)) ) ).

% ATP.lambda_48
tff(fact_10224_ATP_Olambda__49,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ml,Uu),Uua) = $let(
        m2: int,
        m2:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Uu)),
        $ite(Uua = zero_zero_nat,m2,aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),one_one_int)) ) ).

% ATP.lambda_49
tff(fact_10225_ATP_Olambda__50,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_aqm(Uu),Uua)
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = aa_real_complex(cis,Uua) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi) ) ) ).

% ATP.lambda_50
tff(fact_10226_ATP_Olambda__51,axiom,
    ! [Uu: real,Uua: int] :
      ( aa_int_o(aa_real_fun_int_o(aTP_Lamm_aqa,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uu),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_51
tff(fact_10227_ATP_Olambda__52,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_aqb(Uu),Uua)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uu),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_52
tff(fact_10228_ATP_Olambda__53,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ul,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_53
tff(fact_10229_ATP_Olambda__54,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_atc,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_54
tff(fact_10230_ATP_Olambda__55,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ath(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_55
tff(fact_10231_ATP_Olambda__56,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aoy,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% ATP.lambda_56
tff(fact_10232_ATP_Olambda__57,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aoq,Uu),Uua) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_57
tff(fact_10233_ATP_Olambda__58,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_auf,Uu),Uua) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_58
tff(fact_10234_ATP_Olambda__59,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aou,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uu) ).

% ATP.lambda_59
tff(fact_10235_ATP_Olambda__60,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aot,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uua) ).

% ATP.lambda_60
tff(fact_10236_ATP_Olambda__61,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_ach,Uu),Uua)
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_61
tff(fact_10237_ATP_Olambda__62,axiom,
    ! [Uu: nat,Uua: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_acg,Uu),Uua)
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_62
tff(fact_10238_ATP_Olambda__63,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_atr,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(divide_divide_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_63
tff(fact_10239_ATP_Olambda__64,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_aug(Uu),Uua) = aa_real_real(powr_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(divide_divide_real(Uu),Uua))),Uua) ).

% ATP.lambda_64
tff(fact_10240_ATP_Olambda__65,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uk,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_65
tff(fact_10241_ATP_Olambda__66,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_apx,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi)
        & ( aa_real_real(cos_real,Uua) = Uu ) ) ) ).

% ATP.lambda_66
tff(fact_10242_ATP_Olambda__67,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ann(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_67
tff(fact_10243_ATP_Olambda__68,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_anl(Uu),Uua) = minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua)),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_68
tff(fact_10244_ATP_Olambda__69,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_anm(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,aa_nat_nat(suc,Uua))),aa_nat_int(Uu,Uua)) ).

% ATP.lambda_69
tff(fact_10245_ATP_Olambda__70,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_tu(Uu),Uua) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uua)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),Uua)) ).

% ATP.lambda_70
tff(fact_10246_ATP_Olambda__71,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_tv(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(Uu,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).

% ATP.lambda_71
tff(fact_10247_ATP_Olambda__72,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ahq(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_72
tff(fact_10248_ATP_Olambda__73,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_vu(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)),aa_nat_int(power_power_int(zero_zero_int),Uua)) ).

% ATP.lambda_73
tff(fact_10249_ATP_Olambda__74,axiom,
    ! [Uu: fun_Co3960896333048504454r_real,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_amu(Uu),Uua)
    <=> ( aa_Code_integer_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_74
tff(fact_10250_ATP_Olambda__75,axiom,
    ! [Uu: fun_Code_integer_rat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_amx(Uu),Uua)
    <=> ( aa_Code_integer_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_75
tff(fact_10251_ATP_Olambda__76,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_ana(Uu),Uua)
    <=> ( aa_Code_integer_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_76
tff(fact_10252_ATP_Olambda__77,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_amt(Uu),Uua)
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_77
tff(fact_10253_ATP_Olambda__78,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_amw(Uu),Uua)
    <=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_78
tff(fact_10254_ATP_Olambda__79,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_amz(Uu),Uua)
    <=> ( aa_complex_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_79
tff(fact_10255_ATP_Olambda__80,axiom,
    ! [Uu: fun_complex_int,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_anb(Uu),Uua)
    <=> ( aa_complex_int(Uu,Uua) = zero_zero_int ) ) ).

% ATP.lambda_80
tff(fact_10256_ATP_Olambda__81,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( aa_int_o(aTP_Lamm_ams(Uu),Uua)
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_81
tff(fact_10257_ATP_Olambda__82,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_amv(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_82
tff(fact_10258_ATP_Olambda__83,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_amy(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_83
tff(fact_10259_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_atx(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_ath(Uu),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_84
tff(fact_10260_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_atw(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_ath(Uu),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_85
tff(fact_10261_ATP_Olambda__86,axiom,
    ! [Uu: code_integer,Uua: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aTP_Lamm_asn,Uu),(Uua)) = produc4188289175737317920o_char(aTP_Lamm_asm((Uua)),code_bit_cut_integer(Uu)) ).

% ATP.lambda_86
tff(fact_10262_ATP_Olambda__87,axiom,
    ! [Uu: produc3881548065746020326Ti_nat,Uua: fun_Pr4171499318505651870_VEBTi] : aa_fun943973709508947813_VEBTi(aTP_Lamm_xr(Uu),Uua) = aa_Pro1482084341531967383_VEBTi(produc3255295512018472142_VEBTi(aTP_Lamm_xq(Uua)),Uu) ).

% ATP.lambda_87
tff(fact_10263_ATP_Olambda__88,axiom,
    ! [Uu: produc3881548065746020326Ti_nat,Uua: fun_Pr238758252084143515Heap_o] : aa_fun6848502500920544641Heap_o(aTP_Lamm_sy(Uu),Uua) = aa_Pro3916655985231737292Heap_o(produc770043135277712853Heap_o(aTP_Lamm_sx(Uua)),Uu) ).

% ATP.lambda_88
tff(fact_10264_ATP_Olambda__89,axiom,
    ! [Uu: produc3881548065746020326Ti_nat,Uua: fun_Pr342922516890243977on_nat] : aa_fun7698174670542568229on_nat(aTP_Lamm_si(Uu),Uua) = aa_Pro2606730266749126658on_nat(produc8911080112929139129on_nat(aTP_Lamm_sh(Uua)),Uu) ).

% ATP.lambda_89
tff(fact_10265_ATP_Olambda__90,axiom,
    ! [Uu: produc3881548065746020326Ti_nat,Uua: fun_Pr342922516890243977on_nat] : aa_fun7698174670542568229on_nat(aTP_Lamm_sa(Uu),Uua) = aa_Pro2606730266749126658on_nat(produc8911080112929139129on_nat(aTP_Lamm_rz(Uua)),Uu) ).

% ATP.lambda_90
tff(fact_10266_ATP_Olambda__91,axiom,
    ! [Uu: produc3960855945107176009Ti_nat,Uua: fun_Pr4404027214751104635_VEBTi] : aa_fun7494984083220284744_VEBTi(aTP_Lamm_zt(Uu),Uua) = aa_Pro721367001817565044_VEBTi(produc2943724498215716011_VEBTi(produc3258509807451162059_VEBTi(aTP_Lamm_zs(Uua))),Uu) ).

% ATP.lambda_91
tff(fact_10267_ATP_Olambda__92,axiom,
    ! [Uu: produc3960855945107176009Ti_nat,Uua: fun_Pr940977554174145406Heap_o] : aa_fun3870618101526761188Heap_o(aTP_Lamm_to(Uu),Uua) = aa_Pro6173831900156065711Heap_o(produc5872130906356439992Heap_o(produc844971584604743726Heap_o(aTP_Lamm_tn(Uua))),Uu) ).

% ATP.lambda_92
tff(fact_10268_ATP_Olambda__93,axiom,
    ! [Uu: produc3960855945107176009Ti_nat,Uua: fun_Pr633315466666964198on_nat] : aa_fun1100322450986716680on_nat(aTP_Lamm_pz(Uu),Uua) = aa_Pro2727408364141080927on_nat(produc183673358652719894on_nat(produc3521483158534984758on_nat(aTP_Lamm_py(Uua))),Uu) ).

% ATP.lambda_93
tff(fact_10269_ATP_Olambda__94,axiom,
    ! [Uu: produc3960855945107176009Ti_nat,Uua: fun_Pr633315466666964198on_nat] : aa_fun1100322450986716680on_nat(aTP_Lamm_pi(Uu),Uua) = aa_Pro2727408364141080927on_nat(produc183673358652719894on_nat(produc3521483158534984758on_nat(aTP_Lamm_ph(Uua))),Uu) ).

% ATP.lambda_94
tff(fact_10270_ATP_Olambda__95,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_apk(Uu),Uua) = aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Uu)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Uu))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_95
tff(fact_10271_ATP_Olambda__96,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_aqq,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_96
tff(fact_10272_ATP_Olambda__97,axiom,
    ! [Uu: $o,Uua: nat] : aa_nat_nat(aa_o_fun_nat_nat(aTP_Lamm_ary,(Uu)),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(Uu))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% ATP.lambda_97
tff(fact_10273_ATP_Olambda__98,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aop,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_98
tff(fact_10274_ATP_Olambda__99,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_aon,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_99
tff(fact_10275_ATP_Olambda__100,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_asa,Uu),Uua) = $let(
        l2: num,
        l2:= aa_Code_integer_num(code_num_of_integer,Uu),
        $let(
          l3: num,
          l3:= aa_num_num(aa_num_fun_num_num(plus_plus_num,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_num_num(aa_num_fun_num_num(plus_plus_num,l3),one)) ) ) ).

% ATP.lambda_100
tff(fact_10276_ATP_Olambda__101,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_arz,Uu),Uua) = $let(
        l2: nat,
        l2:= aa_Code_integer_nat(code_nat_of_integer,Uu),
        $let(
          l3: nat,
          l3:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l3),one_one_nat)) ) ) ).

% ATP.lambda_101
tff(fact_10277_ATP_Olambda__102,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_apu,Uu),Uua)
    <=> ( aa_real_real(exp_real,Uua) = Uu ) ) ).

% ATP.lambda_102
tff(fact_10278_ATP_Olambda__103,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_lu,Uu),Uua) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_103
tff(fact_10279_ATP_Olambda__104,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_lv,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)) ).

% ATP.lambda_104
tff(fact_10280_ATP_Olambda__105,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ma,Uu),Uua) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_105
tff(fact_10281_ATP_Olambda__106,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_mb,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_106
tff(fact_10282_ATP_Olambda__107,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aoz,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)) ).

% ATP.lambda_107
tff(fact_10283_ATP_Olambda__108,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ats,Uu),Uua) = aa_real_real(times_times_real(Uu),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_108
tff(fact_10284_ATP_Olambda__109,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_aph,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_o_int(zero_n2684676970156552555ol_int,Uua != zero_zero_int)) ).

% ATP.lambda_109
tff(fact_10285_ATP_Olambda__110,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_atq,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_110
tff(fact_10286_ATP_Olambda__111,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_atn,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_111
tff(fact_10287_ATP_Olambda__112,axiom,
    ! [Uu: nat,Uua: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aTP_Lamm_ns(Uu),Uua) = aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(vEBT_vebt_memberi,Uua),Uu) ).

% ATP.lambda_112
tff(fact_10288_ATP_Olambda__113,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aTP_Lamm_acl(Uu),Uua)
    <=> aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(Uua),Uu) ) ).

% ATP.lambda_113
tff(fact_10289_ATP_Olambda__114,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( aa_set_complex_o(aTP_Lamm_ack(Uu),Uua)
    <=> aa_set_complex_o(ord_le211207098394363844omplex(Uua),Uu) ) ).

% ATP.lambda_114
tff(fact_10290_ATP_Olambda__115,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_acm(Uu),Uua)
    <=> aa_set_nat_o(ord_less_eq_set_nat(Uua),Uu) ) ).

% ATP.lambda_115
tff(fact_10291_ATP_Olambda__116,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aTP_Lamm_acj(Uu),Uua)
    <=> aa_set_int_o(ord_less_eq_set_int(Uua),Uu) ) ).

% ATP.lambda_116
tff(fact_10292_ATP_Olambda__117,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aci,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_117
tff(fact_10293_ATP_Olambda__118,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ark,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_118
tff(fact_10294_ATP_Olambda__119,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_fa,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),Uu) ) ).

% ATP.lambda_119
tff(fact_10295_ATP_Olambda__120,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_fc,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uua),Uu) ) ).

% ATP.lambda_120
tff(fact_10296_ATP_Olambda__121,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_fe,Uu),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Uua),Uu) ) ).

% ATP.lambda_121
tff(fact_10297_ATP_Olambda__122,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_fg,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_122
tff(fact_10298_ATP_Olambda__123,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_fi,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu) ) ).

% ATP.lambda_123
tff(fact_10299_ATP_Olambda__124,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_jg(Uu),Uua) = aa_real_real(times_times_real(Uua),Uu) ).

% ATP.lambda_124
tff(fact_10300_ATP_Olambda__125,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aTP_Lamm_jh(Uu),Uua) = aa_rat_rat(times_times_rat(Uua),Uu) ).

% ATP.lambda_125
tff(fact_10301_ATP_Olambda__126,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ji,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_126
tff(fact_10302_ATP_Olambda__127,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_jj,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uu) ).

% ATP.lambda_127
tff(fact_10303_ATP_Olambda__128,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_arn,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_128
tff(fact_10304_ATP_Olambda__129,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_asp,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_129
tff(fact_10305_ATP_Olambda__130,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_arm(Uu),Uua) = aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(Uua),Uu) ).

% ATP.lambda_130
tff(fact_10306_ATP_Olambda__131,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_arp,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_131
tff(fact_10307_ATP_Olambda__132,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_arl,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_132
tff(fact_10308_ATP_Olambda__133,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_asr(Uu),Uua) = aa_real_real(powr_real(Uua),Uu) ).

% ATP.lambda_133
tff(fact_10309_ATP_Olambda__134,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_nb,Uu),Uua)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),Uu) ) ).

% ATP.lambda_134
tff(fact_10310_ATP_Olambda__135,axiom,
    ! [Uu: complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_mx(Uu),Uua)
    <=> aa_complex_o(dvd_dvd_complex(Uua),Uu) ) ).

% ATP.lambda_135
tff(fact_10311_ATP_Olambda__136,axiom,
    ! [Uu: uint32,Uua: uint32] :
      ( aa_uint32_o(aTP_Lamm_mz(Uu),Uua)
    <=> aa_uint32_o(dvd_dvd_uint32(Uua),Uu) ) ).

% ATP.lambda_136
tff(fact_10312_ATP_Olambda__137,axiom,
    ! [Uu: word_N3645301735248828278l_num1,Uua: word_N3645301735248828278l_num1] :
      ( aa_wor4452366487969921764num1_o(aTP_Lamm_na(Uu),Uua)
    <=> aa_wor4452366487969921764num1_o(dvd_dv6812691276156420380l_num1(Uua),Uu) ) ).

% ATP.lambda_137
tff(fact_10313_ATP_Olambda__138,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_nc,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_138
tff(fact_10314_ATP_Olambda__139,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_my,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_139
tff(fact_10315_ATP_Olambda__140,axiom,
    ! [Uu: vEBT_VEBT,Uua: real] : aa_rea2776067176874379751T_VEBT(aTP_Lamm_afu(Uu),Uua) = produc6931449550656315951T_VEBT(Uua,Uu) ).

% ATP.lambda_140
tff(fact_10316_ATP_Olambda__141,axiom,
    ! [Uu: vEBT_VEBT,Uua: $o] : aa_o_P4714545900465707367T_VEBT(aTP_Lamm_afv(Uu),(Uua)) = produc2982872950893828659T_VEBT((Uua),Uu) ).

% ATP.lambda_141
tff(fact_10317_ATP_Olambda__142,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat] : aa_nat3316634782163766759T_VEBT(aTP_Lamm_afw(Uu),Uua) = produc599794634098209291T_VEBT(Uua,Uu) ).

% ATP.lambda_142
tff(fact_10318_ATP_Olambda__143,axiom,
    ! [Uu: vEBT_VEBT,Uua: int] : aa_int4520187544020889831T_VEBT(aTP_Lamm_afx(Uu),Uua) = produc3329399203697025711T_VEBT(Uua,Uu) ).

% ATP.lambda_143
tff(fact_10319_ATP_Olambda__144,axiom,
    ! [Uu: produc4813437837504472865T_VEBT,Uua: nat] : aa_nat7182940012017014868T_VEBT(aTP_Lamm_ms(Uu),Uua) = aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(Uua),Uu) ).

% ATP.lambda_144
tff(fact_10320_ATP_Olambda__145,axiom,
    ! [Uu: assn,Uua: assn] : aa_ass5585057816575862475n_assn(aTP_Lamm_mw(Uu),Uua) = aa_ass5585057816575862475n_assn(produc118845697133431529n_assn(Uua),Uu) ).

% ATP.lambda_145
tff(fact_10321_ATP_Olambda__146,axiom,
    ! [Uu: num,Uua: num] : aa_num3156681643174609981um_num(aTP_Lamm_mr(Uu),Uua) = aa_num3156681643174609981um_num(product_Pair_num_num(Uua),Uu) ).

% ATP.lambda_146
tff(fact_10322_ATP_Olambda__147,axiom,
    ! [Uu: num,Uua: nat] : aa_nat8311465100035950781at_num(aTP_Lamm_mt(Uu),Uua) = aa_num5026351239867140595at_num(product_Pair_nat_num(Uua),Uu) ).

% ATP.lambda_147
tff(fact_10323_ATP_Olambda__148,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_mu,Uu),Uua) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uu) ).

% ATP.lambda_148
tff(fact_10324_ATP_Olambda__149,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_mv,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uu) ).

% ATP.lambda_149
tff(fact_10325_ATP_Olambda__150,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aos,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_150
tff(fact_10326_ATP_Olambda__151,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_atm,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_151
tff(fact_10327_ATP_Olambda__152,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_cc(Uu),Uua)
    <=> aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uua),Uu) ) ).

% ATP.lambda_152
tff(fact_10328_ATP_Olambda__153,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ch(Uu),Uua)
    <=> member5262025264175285858nt_int(Uua,Uu) ) ).

% ATP.lambda_153
tff(fact_10329_ATP_Olambda__154,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_cg(Uu),Uua)
    <=> aa_set_complex_o(member_complex(Uua),Uu) ) ).

% ATP.lambda_154
tff(fact_10330_ATP_Olambda__155,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_cd(Uu),Uua)
    <=> aa_set_real_o(member_real(Uua),Uu) ) ).

% ATP.lambda_155
tff(fact_10331_ATP_Olambda__156,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ce(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_156
tff(fact_10332_ATP_Olambda__157,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_cf(Uu),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_157
tff(fact_10333_ATP_Olambda__158,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_kq(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_158
tff(fact_10334_ATP_Olambda__159,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ks(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_159
tff(fact_10335_ATP_Olambda__160,axiom,
    ! [Uu: complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_kr(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_160
tff(fact_10336_ATP_Olambda__161,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_kt,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_161
tff(fact_10337_ATP_Olambda__162,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ku,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_162
tff(fact_10338_ATP_Olambda__163,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_kv,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_163
tff(fact_10339_ATP_Olambda__164,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_adu(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(Uu,Uua)) ) ).

% ATP.lambda_164
tff(fact_10340_ATP_Olambda__165,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_adm(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Pro6801748904974264397nt_nat(Uu,Uua)) ) ).

% ATP.lambda_165
tff(fact_10341_ATP_Olambda__166,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ads(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(Uu,Uua)) ) ).

% ATP.lambda_166
tff(fact_10342_ATP_Olambda__167,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ado(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_167
tff(fact_10343_ATP_Olambda__168,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_adq(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(Uu,Uua)) ) ).

% ATP.lambda_168
tff(fact_10344_ATP_Olambda__169,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aoj(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_169
tff(fact_10345_ATP_Olambda__170,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_aoi(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_170
tff(fact_10346_ATP_Olambda__171,axiom,
    ! [Uu: fun_fu1421233379579251956EBTi_o,Uua: fun_Pr4404027214751104635_VEBTi] :
      ( aa_fun2613385842899212575EBTi_o(aTP_Lamm_wa(Uu),Uua)
    <=> aa_fun2485637723310249309EBTi_o(Uu,produc3311654011758026858_VEBTi(produc2164094337957399884_VEBTi(Uua))) ) ).

% ATP.lambda_171
tff(fact_10347_ATP_Olambda__172,axiom,
    ! [Uu: fun_fu6711510597495717193_nat_o,Uua: fun_Pr633315466666964198on_nat] :
      ( aa_fun8910613404238449780_nat_o(aTP_Lamm_ro(Uu),Uua)
    <=> aa_fun5369152942363543986_nat_o(Uu,produc9121605208400662357on_nat(produc1757988346207259447on_nat(Uua))) ) ).

% ATP.lambda_172
tff(fact_10348_ATP_Olambda__173,axiom,
    ! [Uu: fun_fu2707964056798801139ap_o_o,Uua: fun_Pr940977554174145406Heap_o] :
      ( aa_fun3656762788152972374ap_o_o(aTP_Lamm_tw(Uu),Uua)
    <=> aa_fun339294776471787884ap_o_o(Uu,produc8906325494159015119Heap_o(produc8381543706267210711Heap_o(Uua))) ) ).

% ATP.lambda_173
tff(fact_10349_ATP_Olambda__174,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_aso(Uu),Uua) = aa_real_real(Uu,aa_real_real(uminus_uminus_real,Uua)) ).

% ATP.lambda_174
tff(fact_10350_ATP_Olambda__175,axiom,
    ! [Uu: fun_na1476630412504255114at_nat,Uua: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aTP_Lamm_hy(Uu),Uua) = aa_nat7603221163087113075at_nat(Uu,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uua)) ).

% ATP.lambda_175
tff(fact_10351_ATP_Olambda__176,axiom,
    ! [Uu: fun_nat_o,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_ia(Uu),Uua)
    <=> aa_nat_o(Uu,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uua)) ) ).

% ATP.lambda_176
tff(fact_10352_ATP_Olambda__177,axiom,
    ! [Uu: fun_in4211310926729683942nt_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_ic(Uu),Uua) = aa_int7462565939943104719nt_int(Uu,aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua)) ).

% ATP.lambda_177
tff(fact_10353_ATP_Olambda__178,axiom,
    ! [Uu: fun_int_o,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ie(Uu),Uua)
    <=> aa_int_o(Uu,aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua)) ) ).

% ATP.lambda_178
tff(fact_10354_ATP_Olambda__179,axiom,
    ! [Uu: fun_int_int,Uua: product_prod_int_int] : aa_Pro6799258434465214121nt_int(aTP_Lamm_ig(Uu),Uua) = aa_int_int(Uu,aa_Pro6799258434465214121nt_int(product_snd_int_int,Uua)) ).

% ATP.lambda_179
tff(fact_10355_ATP_Olambda__180,axiom,
    ! [Uu: fun_na1476630412504255114at_nat,Uua: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aTP_Lamm_ho(Uu),Uua) = aa_nat7603221163087113075at_nat(Uu,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uua)) ).

% ATP.lambda_180
tff(fact_10356_ATP_Olambda__181,axiom,
    ! [Uu: fun_nat_o,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_hq(Uu),Uua)
    <=> aa_nat_o(Uu,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uua)) ) ).

% ATP.lambda_181
tff(fact_10357_ATP_Olambda__182,axiom,
    ! [Uu: fun_in4211310926729683942nt_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aTP_Lamm_hs(Uu),Uua) = aa_int7462565939943104719nt_int(Uu,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua)) ).

% ATP.lambda_182
tff(fact_10358_ATP_Olambda__183,axiom,
    ! [Uu: fun_int_o,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_hu(Uu),Uua)
    <=> aa_int_o(Uu,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua)) ) ).

% ATP.lambda_183
tff(fact_10359_ATP_Olambda__184,axiom,
    ! [Uu: fun_int_int,Uua: product_prod_int_int] : aa_Pro6799258434465214121nt_int(aTP_Lamm_hw(Uu),Uua) = aa_int_int(Uu,aa_Pro6799258434465214121nt_int(product_fst_int_int,Uua)) ).

% ATP.lambda_184
tff(fact_10360_ATP_Olambda__185,axiom,
    ! [Uu: fun_fu1178290271985267629EBTi_o,Uua: fun_Pr4171499318505651870_VEBTi] :
      ( aa_fun4096973365178274620EBTi_o(aTP_Lamm_ys(Uu),Uua)
    <=> aa_fun6333388201673717798EBTi_o(Uu,produc5159149307777246319_VEBTi(Uua)) ) ).

% ATP.lambda_185
tff(fact_10361_ATP_Olambda__186,axiom,
    ! [Uu: fun_fu4750890421531948930_nat_o,Uua: fun_Pr342922516890243977on_nat] :
      ( aa_fun7247719965521315857_nat_o(aTP_Lamm_tp(Uu),Uua)
    <=> aa_fun7244059194882734075_nat_o(Uu,produc1489253303066280154on_nat(Uua)) ) ).

% ATP.lambda_186
tff(fact_10362_ATP_Olambda__187,axiom,
    ! [Uu: fun_fu8886734894937297424ap_o_o,Uua: fun_Pr238758252084143515Heap_o] :
      ( aa_fun1852402840713402361ap_o_o(aTP_Lamm_ub(Uu),Uua)
    <=> aa_fun4656092467790859897ap_o_o(Uu,produc5685940877448195828Heap_o(Uua)) ) ).

% ATP.lambda_187
tff(fact_10363_ATP_Olambda__188,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_vm(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_188
tff(fact_10364_ATP_Olambda__189,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vl(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_189
tff(fact_10365_ATP_Olambda__190,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_arx(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_190
tff(fact_10366_ATP_Olambda__191,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_anj(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_191
tff(fact_10367_ATP_Olambda__192,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_amg(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_192
tff(fact_10368_ATP_Olambda__193,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ank(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_193
tff(fact_10369_ATP_Olambda__194,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_apq(Uu),Uua) = aa_num_option_num(some_num,aa_option_num_num(case_option_num_num(one,bit1),aa_num_option_num(bit_take_bit_num(Uu),Uua))) ).

% ATP.lambda_194
tff(fact_10370_ATP_Olambda__195,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_apm(Uu),Uua) = aa_num_option_num(some_num,aa_option_num_num(case_option_num_num(one,bit1),aa_num_option_num(bit_take_bit_num(Uua),Uu))) ).

% ATP.lambda_195
tff(fact_10371_ATP_Olambda__196,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_atf(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ate(Uu),Uua)) ).

% ATP.lambda_196
tff(fact_10372_ATP_Olambda__197,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_apf,Uu),Uua) = aa_real_complex(cis,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_197
tff(fact_10373_ATP_Olambda__198,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_auc(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) != zero_zero_real ) ) ).

% ATP.lambda_198
tff(fact_10374_ATP_Olambda__199,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_apg,Uu),Uua)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ) ).

% ATP.lambda_199
tff(fact_10375_ATP_Olambda__200,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_atp,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_200
tff(fact_10376_ATP_Olambda__201,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ars,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_201
tff(fact_10377_ATP_Olambda__202,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_arr,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_202
tff(fact_10378_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_apw,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_203
tff(fact_10379_ATP_Olambda__204,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_apv,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_204
tff(fact_10380_ATP_Olambda__205,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_real(aTP_Lamm_aig(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_205
tff(fact_10381_ATP_Olambda__206,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_uy(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_206
tff(fact_10382_ATP_Olambda__207,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ux(Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_207
tff(fact_10383_ATP_Olambda__208,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_atj(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_208
tff(fact_10384_ATP_Olambda__209,axiom,
    ! [Uu: fun_VEBT_VEBT_nat,Uua: vEBT_VEBT] : aa_VEBT_VEBT_real(aTP_Lamm_mp(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(Uu,Uua)) ).

% ATP.lambda_209
tff(fact_10385_ATP_Olambda__210,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_agi(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_210
tff(fact_10386_ATP_Olambda__211,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_agl(Uu),Uua) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_211
tff(fact_10387_ATP_Olambda__212,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_mq(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_212
tff(fact_10388_ATP_Olambda__213,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_agm(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_213
tff(fact_10389_ATP_Olambda__214,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_agk(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_214
tff(fact_10390_ATP_Olambda__215,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_agj(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_215
tff(fact_10391_ATP_Olambda__216,axiom,
    ! [Uu: fun_complex_real,Uua: complex] : aa_complex_complex(aTP_Lamm_agu(Uu),Uua) = aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(Uu,Uua)) ).

% ATP.lambda_216
tff(fact_10392_ATP_Olambda__217,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_complex(aTP_Lamm_agg(Uu),Uua) = aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_217
tff(fact_10393_ATP_Olambda__218,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_agv(Uu),Uua) = aa_real_real(real_V1803761363581548252l_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_218
tff(fact_10394_ATP_Olambda__219,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_complex(aTP_Lamm_aja(Uu),Uua) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_219
tff(fact_10395_ATP_Olambda__220,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_vn(Uu),Uua) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_220
tff(fact_10396_ATP_Olambda__221,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vo(Uu),Uua) = aa_real_real(uminus_uminus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_221
tff(fact_10397_ATP_Olambda__222,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_ajb(Uu),Uua) = aa_int_int(uminus_uminus_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_222
tff(fact_10398_ATP_Olambda__223,axiom,
    ! [Uu: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_agn(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu,Uua)) ).

% ATP.lambda_223
tff(fact_10399_ATP_Olambda__224,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_ago(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_224
tff(fact_10400_ATP_Olambda__225,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int2304959211779887584l_num1(aTP_Lamm_agr(Uu),Uua) = ring_17408606157368542149l_num1(aa_int_int(Uu,Uua)) ).

% ATP.lambda_225
tff(fact_10401_ATP_Olambda__226,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_agp(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu,Uua)) ).

% ATP.lambda_226
tff(fact_10402_ATP_Olambda__227,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_agq(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ).

% ATP.lambda_227
tff(fact_10403_ATP_Olambda__228,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_ags(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_228
tff(fact_10404_ATP_Olambda__229,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vp(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_229
tff(fact_10405_ATP_Olambda__230,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_agt(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_230
tff(fact_10406_ATP_Olambda__231,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_atv(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_231
tff(fact_10407_ATP_Olambda__232,axiom,
    ! [Uu: fun_VE6440932821983644738omplex,Uua: vEBT_VEBT] : aa_VEBT_VEBT_complex(aTP_Lamm_amm(Uu),Uua) = suminf_complex(aa_VEB6050840761258413811omplex(Uu,Uua)) ).

% ATP.lambda_232
tff(fact_10408_ATP_Olambda__233,axiom,
    ! [Uu: fun_VE2989161857680785728t_real,Uua: vEBT_VEBT] : aa_VEBT_VEBT_real(aTP_Lamm_ami(Uu),Uua) = suminf_real(aa_VEB3736018289492012273t_real(Uu,Uua)) ).

% ATP.lambda_233
tff(fact_10409_ATP_Olambda__234,axiom,
    ! [Uu: fun_co3258106519541737638omplex,Uua: complex] : aa_complex_complex(aTP_Lamm_amq(Uu),Uua) = suminf_complex(aa_com8181001469867462503omplex(Uu,Uua)) ).

% ATP.lambda_234
tff(fact_10410_ATP_Olambda__235,axiom,
    ! [Uu: fun_re2084383175013312036omplex,Uua: real] : aa_real_complex(aTP_Lamm_amo(Uu),Uua) = suminf_complex(aa_rea1531280491672885477omplex(Uu,Uua)) ).

% ATP.lambda_235
tff(fact_10411_ATP_Olambda__236,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_amk(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_236
tff(fact_10412_ATP_Olambda__237,axiom,
    ! [Uu: fun_na1142814060743658952omplex,Uua: nat] : aa_nat_complex(aTP_Lamm_aml(Uu),Uua) = suminf_complex(aa_nat6454746962644518025omplex(Uu,Uua)) ).

% ATP.lambda_237
tff(fact_10413_ATP_Olambda__238,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_amr(Uu),Uua) = suminf_real(aa_nat_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_238
tff(fact_10414_ATP_Olambda__239,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_amp(Uu),Uua) = suminf_nat(aa_nat_fun_nat_nat(Uu,Uua)) ).

% ATP.lambda_239
tff(fact_10415_ATP_Olambda__240,axiom,
    ! [Uu: fun_in8056368165505483812omplex,Uua: int] : aa_int_complex(aTP_Lamm_amn(Uu),Uua) = suminf_complex(aa_int4144929030551567077omplex(Uu,Uua)) ).

% ATP.lambda_240
tff(fact_10416_ATP_Olambda__241,axiom,
    ! [Uu: fun_int_fun_nat_real,Uua: int] : aa_int_real(aTP_Lamm_amj(Uu),Uua) = suminf_real(aa_int_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_241
tff(fact_10417_ATP_Olambda__242,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_aru(Uu),Uua)
    <=> ! [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),M6)
         => ( aa_nat_o(Uu,M6)
          <=> aa_nat_o(Uu,Uua) ) ) ) ).

% ATP.lambda_242
tff(fact_10418_ATP_Olambda__243,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_arw(Uu),Uua)
    <=> ! [N11: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),N11)
         => aa_nat_o(Uu,N11) ) ) ).

% ATP.lambda_243
tff(fact_10419_ATP_Olambda__244,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_arv(Uu),Uua)
    <=> ! [N11: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),N11)
         => ~ aa_nat_o(Uu,N11) ) ) ).

% ATP.lambda_244
tff(fact_10420_ATP_Olambda__245,axiom,
    ! [Uu: vEBT_VEBTi,Uua: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aTP_Lamm_yt,Uu),Uua) = heap_T1489671670754571048_VEBTi(aTP_Lamm_sp) ).

% ATP.lambda_245
tff(fact_10421_ATP_Olambda__246,axiom,
    ! [Uu: vEBT_VEBTi,Uua: nat] : aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aTP_Lamm_uc,Uu),Uua) = heap_Time_Heap_o2(aTP_Lamm_so) ).

% ATP.lambda_246
tff(fact_10422_ATP_Olambda__247,axiom,
    ! [Uu: vEBT_VEBTi,Uua: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aTP_Lamm_tq,Uu),Uua) = heap_T5286843759275942675on_nat(aTP_Lamm_rp) ).

% ATP.lambda_247
tff(fact_10423_ATP_Olambda__248,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: vEBT_VEBTi,Uub: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aTP_Lamm_yu(Uu),Uua),Uub) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_nat3009484633486537743_VEBTi(aTP_Lamm_ye(Uu),Uub),aTP_Lamm_xb(Uub)),Uua) ).

% ATP.lambda_248
tff(fact_10424_ATP_Olambda__249,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: vEBT_VEBTi,Uub: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aTP_Lamm_xq(Uu),Uua),Uub) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_nat3009484633486537743_VEBTi(aTP_Lamm_xp(Uu),Uub),aTP_Lamm_xb(Uub)),Uua) ).

% ATP.lambda_249
tff(fact_10425_ATP_Olambda__250,axiom,
    ! [Uu: fun_Pr238758252084143515Heap_o,Uua: vEBT_VEBTi,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aTP_Lamm_sx(Uu),Uua),Uub) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_nat1227502447850154924Heap_o(aTP_Lamm_sw(Uu),Uub),aTP_Lamm_nr(Uub)),Uua) ).

% ATP.lambda_250
tff(fact_10426_ATP_Olambda__251,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: vEBT_VEBTi,Uub: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aTP_Lamm_sh(Uu),Uua),Uub) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_sg(Uu),Uub),aTP_Lamm_bk(Uub)),Uua) ).

% ATP.lambda_251
tff(fact_10427_ATP_Olambda__252,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: vEBT_VEBTi,Uub: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aTP_Lamm_rz(Uu),Uua),Uub) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_ry(Uu),Uub),aTP_Lamm_cq(Uub)),Uua) ).

% ATP.lambda_252
tff(fact_10428_ATP_Olambda__253,axiom,
    ! [Uu: fun_VE7134199479003659425Heap_o,Uua: vEBT_VEBTi,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aTP_Lamm_ud(Uu),Uua),Uub) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_nat1227502447850154924Heap_o(aTP_Lamm_or(Uu),Uub),aTP_Lamm_nr(Uub)),Uua) ).

% ATP.lambda_253
tff(fact_10429_ATP_Olambda__254,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: vEBT_VEBTi,Uub: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aTP_Lamm_ts(Uu),Uua),Uub) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_hj(Uu),Uub),aTP_Lamm_cq(Uub)),Uua) ).

% ATP.lambda_254
tff(fact_10430_ATP_Olambda__255,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: vEBT_VEBTi,Uub: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aTP_Lamm_tr(Uu),Uua),Uub) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_hd(Uu),Uub),aTP_Lamm_bk(Uub)),Uua) ).

% ATP.lambda_255
tff(fact_10431_ATP_Olambda__256,axiom,
    ! [Uu: nat,Uua: $o,Uub: $o] :
      aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(aTP_Lamm_bk(Uu),(Uua)),(Uub)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu),
        $ite(
          (Uub),
          heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),
          $ite((Uua),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)) ),
        $ite(
          Uu = one_one_nat,
          $ite((Uua),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)),
          heap_T3487192422709364219on_nat(none_nat) ) ) ).

% ATP.lambda_256
tff(fact_10432_ATP_Olambda__257,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_aog(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_257
tff(fact_10433_ATP_Olambda__258,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_aoh(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uu,Uub),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_258
tff(fact_10434_ATP_Olambda__259,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_jd(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),Uub),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uub),numera6620942414471956472nteger(Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_259
tff(fact_10435_ATP_Olambda__260,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_jc(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu))),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_260
tff(fact_10436_ATP_Olambda__261,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_jb(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_261
tff(fact_10437_ATP_Olambda__262,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_abc(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_262
tff(fact_10438_ATP_Olambda__263,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_abd(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_263
tff(fact_10439_ATP_Olambda__264,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_abe(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_264
tff(fact_10440_ATP_Olambda__265,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_abf(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_265
tff(fact_10441_ATP_Olambda__266,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_agw(Uu),Uua),Uub) = $ite(Uu = Uub,aa_VEBT_VEBT_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_266
tff(fact_10442_ATP_Olambda__267,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_ahb(Uu),Uua),Uub) = $ite(Uu = Uub,aa_VEBT_VEBT_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_267
tff(fact_10443_ATP_Olambda__268,axiom,
    ! [Uu: code_integer,Uua: fun_Co3960896333048504454r_real,Uub: code_integer] :
      aa_Code_integer_real(aa_fun6341089502340155082r_real(aTP_Lamm_aha(Uu),Uua),Uub) = $ite(Uu = Uub,aa_Code_integer_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_268
tff(fact_10444_ATP_Olambda__269,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_agz(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_269
tff(fact_10445_ATP_Olambda__270,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_ahf(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_270
tff(fact_10446_ATP_Olambda__271,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_agx(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_271
tff(fact_10447_ATP_Olambda__272,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_ahc(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_272
tff(fact_10448_ATP_Olambda__273,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ahd(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_273
tff(fact_10449_ATP_Olambda__274,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_agy(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_274
tff(fact_10450_ATP_Olambda__275,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ahe(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_275
tff(fact_10451_ATP_Olambda__276,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aTP_Lamm_ahg(Uu),Uua),Uub) = $ite(Uub = Uu,aa_VEBT_VEBT_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_276
tff(fact_10452_ATP_Olambda__277,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aTP_Lamm_ahl(Uu),Uua),Uub) = $ite(Uub = Uu,aa_VEBT_VEBT_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_277
tff(fact_10453_ATP_Olambda__278,axiom,
    ! [Uu: code_integer,Uua: fun_Co3960896333048504454r_real,Uub: code_integer] :
      aa_Code_integer_real(aa_fun6341089502340155082r_real(aTP_Lamm_ahk(Uu),Uua),Uub) = $ite(Uub = Uu,aa_Code_integer_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_278
tff(fact_10454_ATP_Olambda__279,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_ahj(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_279
tff(fact_10455_ATP_Olambda__280,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_ahp(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_280
tff(fact_10456_ATP_Olambda__281,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ahh(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_281
tff(fact_10457_ATP_Olambda__282,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_ahm(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_282
tff(fact_10458_ATP_Olambda__283,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_un(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_283
tff(fact_10459_ATP_Olambda__284,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_uo(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_284
tff(fact_10460_ATP_Olambda__285,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ahn(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_285
tff(fact_10461_ATP_Olambda__286,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_up(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_286
tff(fact_10462_ATP_Olambda__287,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_uq(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_287
tff(fact_10463_ATP_Olambda__288,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ahi(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_288
tff(fact_10464_ATP_Olambda__289,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_aho(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_289
tff(fact_10465_ATP_Olambda__290,axiom,
    ! [Uu: nat,Uua: $o,Uub: $o] :
      aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(aTP_Lamm_cq(Uu),(Uua)),(Uub)) = $ite(
        Uu = zero_zero_nat,
        $ite((Uub),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,one_one_nat)),heap_T3487192422709364219on_nat(none_nat)),
        heap_T3487192422709364219on_nat(none_nat) ) ).

% ATP.lambda_290
tff(fact_10466_ATP_Olambda__291,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ase(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu)),Uub))) ).

% ATP.lambda_291
tff(fact_10467_ATP_Olambda__292,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_asc(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Uu)),Uub))) ).

% ATP.lambda_292
tff(fact_10468_ATP_Olambda__293,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_asd(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(Uu),Uub))) ).

% ATP.lambda_293
tff(fact_10469_ATP_Olambda__294,axiom,
    ! [Uu: nat,Uua: $o,Uub: $o] :
      aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(aTP_Lamm_xb(Uu),(Uua)),(Uub)) = $ite(
        Uu = zero_zero_nat,
        aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi($true,(Uub))),
        $ite(Uu = one_one_nat,aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi((Uua),$true)),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,vEBT_Leafi((Uua),(Uub)))) ) ).

% ATP.lambda_294
tff(fact_10470_ATP_Olambda__295,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: option_nat] :
      aa_opt9163293676379396302on_nat(aa_arr3757929795460847279on_nat(aTP_Lamm_cj(Uu),Uua),Uub) = $ite(Uub = none_nat,heap_T3487192422709364219on_nat(none_nat),heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),the_nat(Uub)),aa_opt1214731524852023858on_nat(aTP_Lamm_ci(Uu),Uub))) ).

% ATP.lambda_295
tff(fact_10471_ATP_Olambda__296,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_abg(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_296
tff(fact_10472_ATP_Olambda__297,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_abh(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_297
tff(fact_10473_ATP_Olambda__298,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_abi(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_298
tff(fact_10474_ATP_Olambda__299,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_abj(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_299
tff(fact_10475_ATP_Olambda__300,axiom,
    ! [Uu: fun_VEBT_VEBT_real,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_real(aa_fun3597767889293385422T_real(aTP_Lamm_alr(Uu),Uua),Uub) = $ite(aa_VEBT_VEBT_o(Uua,Uub),aa_VEBT_VEBT_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_300
tff(fact_10476_ATP_Olambda__301,axiom,
    ! [Uu: fun_VEBT_VEBT_rat,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      aa_VEBT_VEBT_rat(aa_fun7833968636320851834BT_rat(aTP_Lamm_ama(Uu),Uua),Uub) = $ite(aa_VEBT_VEBT_o(Uua,Uub),aa_VEBT_VEBT_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_301
tff(fact_10477_ATP_Olambda__302,axiom,
    ! [Uu: fun_Co3960896333048504454r_real,Uua: fun_Code_integer_o,Uub: code_integer] :
      aa_Code_integer_real(aa_fun7534015886319449118r_real(aTP_Lamm_alz(Uu),Uua),Uub) = $ite(aa_Code_integer_o(Uua,Uub),aa_Code_integer_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_302
tff(fact_10478_ATP_Olambda__303,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_o,Uub: complex] :
      aa_complex_real(aa_fun8261057609414086622x_real(aTP_Lamm_alx(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_303
tff(fact_10479_ATP_Olambda__304,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_o,Uub: complex] :
      aa_complex_rat(aa_fun4010016270681809034ex_rat(aTP_Lamm_amf(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_304
tff(fact_10480_ATP_Olambda__305,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_o,Uub: real] :
      aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_alt(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_305
tff(fact_10481_ATP_Olambda__306,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_o,Uub: real] :
      aa_real_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_amb(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_306
tff(fact_10482_ATP_Olambda__307,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_amd(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_307
tff(fact_10483_ATP_Olambda__308,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_o,Uub: int] :
      aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_alv(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_308
tff(fact_10484_ATP_Olambda__309,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] :
      aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_ame(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_309
tff(fact_10485_ATP_Olambda__310,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_atz(Uu),Uua),Uub)
    <=> has_fi5821293074295781190e_real(Uu,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_310
tff(fact_10486_ATP_Olambda__311,axiom,
    ! [Uu: fun_Pr238758252084143515Heap_o,Uua: nat,Uub: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_nat1164146675577659234Heap_o(aTP_Lamm_sr(Uu),Uua),Uub) = aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(produc5685940877448195828Heap_o(Uu),Uub),Uua) ).

% ATP.lambda_311
tff(fact_10487_ATP_Olambda__312,axiom,
    ! [Uu: fun_VE7134199479003659425Heap_o,Uua: nat,Uub: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_nat1164146675577659234Heap_o(aTP_Lamm_om(Uu),Uua),Uub) = aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_312
tff(fact_10488_ATP_Olambda__313,axiom,
    ! [Uu: fun_VE6491116402277797979_VEBTi,Uua: fun_Pr238758252084143515Heap_o,Uub: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_fun8469889001190476422_VEBTi(aTP_Lamm_qm(Uu),Uua),Uub) = aa_fun238413853627933730_VEBTi(aa_VEB5816640871196325404_VEBTi(Uu,Uub),Uua) ).

% ATP.lambda_313
tff(fact_10489_ATP_Olambda__314,axiom,
    ! [Uu: fun_VE6943415934277719092Heap_o,Uua: fun_Pr238758252084143515Heap_o,Uub: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_fun2163459272138508211Heap_o(aTP_Lamm_qq(Uu),Uua),Uub) = aa_fun6848502500920544641Heap_o(aa_VEB6642844428796722333Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_314
tff(fact_10490_ATP_Olambda__315,axiom,
    ! [Uu: fun_VE2006399705683203104omplex,Uua: complex,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_com2862003112617944841omplex(aTP_Lamm_akb(Uu),Uua),Uub) = aa_complex_complex(aa_VEB5608662635949521omplex(Uu,Uub),Uua) ).

% ATP.lambda_315
tff(fact_10491_ATP_Olambda__316,axiom,
    ! [Uu: fun_VE6440932821983644738omplex,Uua: nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_nat4246333384384266855omplex(aTP_Lamm_alg(Uu),Uua),Uub) = aa_nat_complex(aa_VEB6050840761258413811omplex(Uu,Uub),Uua) ).

% ATP.lambda_316
tff(fact_10492_ATP_Olambda__317,axiom,
    ! [Uu: fun_VE2989161857680785728t_real,Uua: nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_real(aa_nat4323386229672267365T_real(aTP_Lamm_aky(Uu),Uua),Uub) = aa_nat_real(aa_VEB3736018289492012273t_real(Uu,Uub),Uua) ).

% ATP.lambda_317
tff(fact_10493_ATP_Olambda__318,axiom,
    ! [Uu: fun_VE597728521808591204at_nat,Uua: nat,Uub: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_aje(Uu),Uua),Uub) = aa_nat_nat(aa_VEB1122092716774696213at_nat(Uu,Uub),Uua) ).

% ATP.lambda_318
tff(fact_10494_ATP_Olambda__319,axiom,
    ! [Uu: fun_Co8905303033605938227omplex,Uua: complex,Uub: code_integer] : aa_Cod5621350732091595593omplex(aa_com7552888802784664052omplex(aTP_Lamm_akt(Uu),Uua),Uub) = aa_complex_complex(aa_Cod3397930797123873268omplex(Uu,Uub),Uua) ).

% ATP.lambda_319
tff(fact_10495_ATP_Olambda__320,axiom,
    ! [Uu: fun_Co3312552843764780407at_nat,Uua: nat,Uub: code_integer] : aa_Code_integer_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_ajw(Uu),Uua),Uub) = aa_nat_nat(aa_Cod8670293052248272184at_nat(Uu,Uub),Uua) ).

% ATP.lambda_320
tff(fact_10496_ATP_Olambda__321,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_ahy(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(Uu,Uub),Uua) ).

% ATP.lambda_321
tff(fact_10497_ATP_Olambda__322,axiom,
    ! [Uu: fun_co3258106519541737638omplex,Uua: nat,Uub: complex] : aa_complex_complex(aa_nat3829984526574172007omplex(aTP_Lamm_aln(Uu),Uua),Uub) = aa_nat_complex(aa_com8181001469867462503omplex(Uu,Uub),Uua) ).

% ATP.lambda_322
tff(fact_10498_ATP_Olambda__323,axiom,
    ! [Uu: fun_co636942507509095880at_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_ajr(Uu),Uua),Uub) = aa_nat_nat(aa_com6383720667567721865at_nat(Uu,Uub),Uua) ).

% ATP.lambda_323
tff(fact_10499_ATP_Olambda__324,axiom,
    ! [Uu: fun_re7208826350674946562omplex,Uua: complex,Uub: real] : aa_real_complex(aa_com4766338864777664963omplex(aTP_Lamm_akf(Uu),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(Uu,Uub),Uua) ).

% ATP.lambda_324
tff(fact_10500_ATP_Olambda__325,axiom,
    ! [Uu: fun_re2084383175013312036omplex,Uua: nat,Uub: real] : aa_real_complex(aa_nat7791439356240649701omplex(aTP_Lamm_alk(Uu),Uua),Uub) = aa_nat_complex(aa_rea1531280491672885477omplex(Uu,Uub),Uua) ).

% ATP.lambda_325
tff(fact_10501_ATP_Olambda__326,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_alc(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_326
tff(fact_10502_ATP_Olambda__327,axiom,
    ! [Uu: fun_real_fun_nat_nat,Uua: nat,Uub: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_aji(Uu),Uua),Uub) = aa_nat_nat(aa_real_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_327
tff(fact_10503_ATP_Olambda__328,axiom,
    ! [Uu: fun_o_2166024663187074044_VEBTi,Uua: fun_Pr342922516890243977on_nat,Uub: $o] : aa_o_H7577119287286566245_VEBTi(aa_fun8426004761281959347_VEBTi(aTP_Lamm_rc(Uu),Uua),(Uub)) = aa_fun765629416642892858_VEBTi(aa_o_f1510940835853032693_VEBTi(Uu,(Uub)),Uua) ).

% ATP.lambda_328
tff(fact_10504_ATP_Olambda__329,axiom,
    ! [Uu: fun_o_5993975905899854163Heap_o,Uua: fun_Pr342922516890243977on_nat,Uub: $o] : aa_o_H7399033255965945470Heap_o(aa_fun7300956949533034054Heap_o(aTP_Lamm_ra(Uu),Uua),(Uub)) = aa_fun4188577930510380905Heap_o(aa_o_f6675407086664040580Heap_o(Uu,(Uub)),Uua) ).

% ATP.lambda_329
tff(fact_10505_ATP_Olambda__330,axiom,
    ! [Uu: fun_o_7171189317104958676_VEBTi,Uua: fun_Pr238758252084143515Heap_o,Uub: $o] : aa_o_H7577119287286566245_VEBTi(aa_fun6586487760926256075_VEBTi(aTP_Lamm_qs(Uu),Uua),(Uub)) = aa_fun238413853627933730_VEBTi(aa_o_f8811364424157995653_VEBTi(Uu,(Uub)),Uua) ).

% ATP.lambda_330
tff(fact_10506_ATP_Olambda__331,axiom,
    ! [Uu: fun_o_2223814739247024763Heap_o,Uua: fun_Pr238758252084143515Heap_o,Uub: $o] : aa_o_H7399033255965945470Heap_o(aa_fun7917812164634591790Heap_o(aTP_Lamm_qu(Uu),Uua),(Uub)) = aa_fun6848502500920544641Heap_o(aa_o_f6784619210364167924Heap_o(Uu,(Uub)),Uua) ).

% ATP.lambda_331
tff(fact_10507_ATP_Olambda__332,axiom,
    ! [Uu: fun_na1591161823130732966omplex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_akj(Uu),Uua),Uub) = aa_complex_complex(aa_nat3829984526574172007omplex(Uu,Uub),Uua) ).

% ATP.lambda_332
tff(fact_10508_ATP_Olambda__333,axiom,
    ! [Uu: fun_na1142814060743658952omplex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ale(Uu),Uua),Uub) = aa_nat_complex(aa_nat6454746962644518025omplex(Uu,Uub),Uua) ).

% ATP.lambda_333
tff(fact_10509_ATP_Olambda__334,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_aib(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_334
tff(fact_10510_ATP_Olambda__335,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ahv(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_335
tff(fact_10511_ATP_Olambda__336,axiom,
    ! [Uu: fun_na7267038673063590618_VEBTi,Uua: fun_Pr342922516890243977on_nat,Uub: nat] : aa_nat5408703056993926195_VEBTi(aa_fun1166700169754672579_VEBTi(aTP_Lamm_qy(Uu),Uua),Uub) = aa_fun765629416642892858_VEBTi(aa_nat5169694059731795907_VEBTi(Uu,Uub),Uua) ).

% ATP.lambda_336
tff(fact_10512_ATP_Olambda__337,axiom,
    ! [Uu: fun_na979242554834862325Heap_o,Uua: fun_Pr342922516890243977on_nat,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_fun1690681901636463158Heap_o(aTP_Lamm_qw(Uu),Uua),Uub) = aa_fun4188577930510380905Heap_o(aa_nat5421218410817480758Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_337
tff(fact_10513_ATP_Olambda__338,axiom,
    ! [Uu: fun_na5845961453008041334_VEBTi,Uua: fun_Pr238758252084143515Heap_o,Uub: nat] : aa_nat5408703056993926195_VEBTi(aa_fun5587972357748661675_VEBTi(aTP_Lamm_qk(Uu),Uua),Uub) = aa_fun238413853627933730_VEBTi(aa_nat4209467000370889527_VEBTi(Uu,Uub),Uua) ).

% ATP.lambda_338
tff(fact_10514_ATP_Olambda__339,axiom,
    ! [Uu: fun_na8517577583218006873Heap_o,Uua: fun_Pr238758252084143515Heap_o,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_fun7995322860690487374Heap_o(aTP_Lamm_qo(Uu),Uua),Uub) = aa_fun6848502500920544641Heap_o(aa_nat205861980368755650Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_339
tff(fact_10515_ATP_Olambda__340,axiom,
    ! [Uu: fun_in1307110748760623106omplex,Uua: complex,Uub: int] : aa_int_complex(aa_com9156536935073639875omplex(aTP_Lamm_ako(Uu),Uua),Uub) = aa_complex_complex(aa_int3545933452204062147omplex(Uu,Uub),Uua) ).

% ATP.lambda_340
tff(fact_10516_ATP_Olambda__341,axiom,
    ! [Uu: fun_in8056368165505483812omplex,Uua: nat,Uub: int] : aa_int_complex(aa_nat7430282427850695397omplex(aTP_Lamm_ali(Uu),Uua),Uub) = aa_nat_complex(aa_int4144929030551567077omplex(Uu,Uub),Uua) ).

% ATP.lambda_341
tff(fact_10517_ATP_Olambda__342,axiom,
    ! [Uu: fun_int_fun_nat_real,Uua: nat,Uub: int] : aa_int_real(aa_nat_fun_int_real(aTP_Lamm_ala(Uu),Uua),Uub) = aa_nat_real(aa_int_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_342
tff(fact_10518_ATP_Olambda__343,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_ajm(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_343
tff(fact_10519_ATP_Olambda__344,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_aie(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_344
tff(fact_10520_ATP_Olambda__345,axiom,
    ! [Uu: fun_fu411293875173739919on_nat,Uua: produc3881548065746020326Ti_nat,Uub: fun_Pr342922516890243977on_nat] : aa_fun7698174670542568229on_nat(aa_Pro8654539709784947602on_nat(aTP_Lamm_uf(Uu),Uua),Uub) = aa_Pro2606730266749126658on_nat(aa_fun943833532922116944on_nat(Uu,Uub),Uua) ).

% ATP.lambda_345
tff(fact_10521_ATP_Olambda__346,axiom,
    ! [Uu: fun_fu2426956696497423235Heap_o,Uua: produc3881548065746020326Ti_nat,Uub: fun_Pr238758252084143515Heap_o] : aa_fun6848502500920544641Heap_o(aa_Pro5165804456746783526Heap_o(aTP_Lamm_ug(Uu),Uua),Uub) = aa_Pro3916655985231737292Heap_o(aa_fun5049358401508434812Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_346
tff(fact_10522_ATP_Olambda__347,axiom,
    ! [Uu: fun_fu1151686773700218227_VEBTi,Uua: produc3960855945107176009Ti_nat,Uub: fun_Pr4404027214751104635_VEBTi] : aa_fun7494984083220284744_VEBTi(aa_Pro7504462699960347282_VEBTi(aTP_Lamm_uh(Uu),Uua),Uub) = aa_Pro721367001817565044_VEBTi(aa_fun5250146550359294516_VEBTi(Uu,Uub),Uua) ).

% ATP.lambda_347
tff(fact_10523_ATP_Olambda__348,axiom,
    ! [Uu: fun_fu3620910142932681on_nat,Uua: produc3960855945107176009Ti_nat,Uub: fun_Pr633315466666964198on_nat] : aa_fun1100322450986716680on_nat(aa_Pro1535628157467025618on_nat(aTP_Lamm_ui(Uu),Uua),Uub) = aa_Pro2727408364141080927on_nat(aa_fun7861206497237518090on_nat(Uu,Uub),Uua) ).

% ATP.lambda_348
tff(fact_10524_ATP_Olambda__349,axiom,
    ! [Uu: fun_fu8308473583445096899Heap_o,Uua: produc3960855945107176009Ti_nat,Uub: fun_Pr940977554174145406Heap_o] : aa_fun3870618101526761188Heap_o(aa_Pro9147008720614807654Heap_o(aTP_Lamm_uj(Uu),Uua),Uub) = aa_Pro6173831900156065711Heap_o(aa_fun55487300827498300Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_349
tff(fact_10525_ATP_Olambda__350,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_asf(Uu),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,aa_Cod1783974652533396728nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),Uub)),Uub = one_one_Code_integer) ).

% ATP.lambda_350
tff(fact_10526_ATP_Olambda__351,axiom,
    ! [Uu: $o,Uua: code_integer,Uub: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aTP_Lamm_asm((Uu)),Uua),(Uub)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aTP_Lamm_asl((Uu)),(Uub)),code_bit_cut_integer(Uua)) ).

% ATP.lambda_351
tff(fact_10527_ATP_Olambda__352,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_arg(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_arf(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_352
tff(fact_10528_ATP_Olambda__353,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_are(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ard(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_353
tff(fact_10529_ATP_Olambda__354,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_arc(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_arb(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_354
tff(fact_10530_ATP_Olambda__355,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ara(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aqz(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_355
tff(fact_10531_ATP_Olambda__356,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aqy(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_aqx(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_356
tff(fact_10532_ATP_Olambda__357,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aqv(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_aqu(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_357
tff(fact_10533_ATP_Olambda__358,axiom,
    ! [Uu: set_complex,Uua: fun_co3258106519541737638omplex,Uub: nat] : aa_nat_complex(aa_fun5003373910569599327omplex(aTP_Lamm_alo(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(aTP_Lamm_aln(Uua),Uub),Uu) ).

% ATP.lambda_358
tff(fact_10534_ATP_Olambda__359,axiom,
    ! [Uu: set_real,Uua: fun_re2084383175013312036omplex,Uub: nat] : aa_nat_complex(aa_fun7632768522510439005omplex(aTP_Lamm_all(Uu),Uua),Uub) = groups5754745047067104278omplex(aa_nat7791439356240649701omplex(aTP_Lamm_alk(Uua),Uub),Uu) ).

% ATP.lambda_359
tff(fact_10535_ATP_Olambda__360,axiom,
    ! [Uu: set_int,Uua: fun_in8056368165505483812omplex,Uub: nat] : aa_nat_complex(aa_fun3543561798013398749omplex(aTP_Lamm_alj(Uu),Uua),Uub) = groups3049146728041665814omplex(aa_nat7430282427850695397omplex(aTP_Lamm_ali(Uua),Uub),Uu) ).

% ATP.lambda_360
tff(fact_10536_ATP_Olambda__361,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE6440932821983644738omplex,Uub: nat] : aa_nat_complex(aa_fun3960763739589619457omplex(aTP_Lamm_alh(Uu),Uua),Uub) = groups1794756597179926696omplex(aa_nat4246333384384266855omplex(aTP_Lamm_alg(Uua),Uub),Uu) ).

% ATP.lambda_361
tff(fact_10537_ATP_Olambda__362,axiom,
    ! [Uu: set_nat,Uua: fun_na1142814060743658952omplex,Uub: nat] : aa_nat_complex(aa_fun4886358607025424513omplex(aTP_Lamm_alf(Uu),Uua),Uub) = groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_ale(Uua),Uub),Uu) ).

% ATP.lambda_362
tff(fact_10538_ATP_Olambda__363,axiom,
    ! [Uu: set_real,Uua: fun_re4238438383274954274t_real,Uub: nat] : aa_nat_real(aa_fun628238740774635481t_real(aTP_Lamm_ald(Uu),Uua),Uub) = groups8097168146408367636l_real(aa_nat_fun_real_real(aTP_Lamm_alc(Uua),Uub),Uu) ).

% ATP.lambda_363
tff(fact_10539_ATP_Olambda__364,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun2842512154077127769t_real(aTP_Lamm_alb(Uu),Uua),Uub) = groups8778361861064173332t_real(aa_nat_fun_int_real(aTP_Lamm_ala(Uua),Uub),Uu) ).

% ATP.lambda_364
tff(fact_10540_ATP_Olambda__365,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE2989161857680785728t_real,Uub: nat] : aa_nat_real(aa_fun6147343916763992189t_real(aTP_Lamm_akz(Uu),Uua),Uub) = groups2240296850493347238T_real(aa_nat4323386229672267365T_real(aTP_Lamm_aky(Uua),Uub),Uu) ).

% ATP.lambda_365
tff(fact_10541_ATP_Olambda__366,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_aif(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_aie(Uu),Uub),Uua) ).

% ATP.lambda_366
tff(fact_10542_ATP_Olambda__367,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: set_nat,Uub: nat] : aa_nat_real(aa_set1166323925324055613t_real(aTP_Lamm_aic(Uu),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_aib(Uu),Uub),Uua) ).

% ATP.lambda_367
tff(fact_10543_ATP_Olambda__368,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_alp(Uu),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_aib(Uua),Uub),Uu) ).

% ATP.lambda_368
tff(fact_10544_ATP_Olambda__369,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: set_complex,Uub: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aTP_Lamm_ahz(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_ahy(Uu),Uub),Uua) ).

% ATP.lambda_369
tff(fact_10545_ATP_Olambda__370,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_ahw(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ahv(Uu),Uub)),Uua) ).

% ATP.lambda_370
tff(fact_10546_ATP_Olambda__371,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_alm(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ahv(Uua),Uub)),Uu) ).

% ATP.lambda_371
tff(fact_10547_ATP_Olambda__372,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_asu(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_372
tff(fact_10548_ATP_Olambda__373,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_asv(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_373
tff(fact_10549_ATP_Olambda__374,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ahr(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),Uub))),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_374
tff(fact_10550_ATP_Olambda__375,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_aht(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_375
tff(fact_10551_ATP_Olambda__376,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_ahs(Uu),Uua),Uub) = divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_376
tff(fact_10552_ATP_Olambda__377,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_atd(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_377
tff(fact_10553_ATP_Olambda__378,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_aom(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_378
tff(fact_10554_ATP_Olambda__379,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_aby(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua) ) ) ).

% ATP.lambda_379
tff(fact_10555_ATP_Olambda__380,axiom,
    ! [Uu: set_Code_integer,Uua: nat,Uub: list_Code_integer] :
      ( aa_lis2813719629158071341eger_o(aa_nat2697030861539126eger_o(aTP_Lamm_acb(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_Code_integer2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3445333598471063425nteger(Uub)),Uua) ) ) ).

% ATP.lambda_380
tff(fact_10556_ATP_Olambda__381,axiom,
    ! [Uu: set_Pr5949110396991348497n_assn,Uua: nat,Uub: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(aa_nat2893684320935325170assn_o(aTP_Lamm_abz(Uu),Uua),Uub)
    <=> ( ord_le171416862856029873n_assn(set_Pr1139785259514867910n_assn(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6829681357464350627n_assn(Uub)),Uua) ) ) ).

% ATP.lambda_381
tff(fact_10557_ATP_Olambda__382,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_aca(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua) ) ) ).

% ATP.lambda_382
tff(fact_10558_ATP_Olambda__383,axiom,
    ! [Uu: set_real,Uua: nat,Uub: list_real] :
      ( aa_list_real_o(aa_nat3629845878562903719real_o(aTP_Lamm_acc(Uu),Uua),Uub)
    <=> ( aa_set_real_o(ord_less_eq_set_real(set_real2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Uub)),Uua) ) ) ).

% ATP.lambda_383
tff(fact_10559_ATP_Olambda__384,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_acd(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_o(set_o2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Uub)),Uua) ) ) ).

% ATP.lambda_384
tff(fact_10560_ATP_Olambda__385,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_acf(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Uub)),Uua) ) ) ).

% ATP.lambda_385
tff(fact_10561_ATP_Olambda__386,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_ace(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Uub)),Uua) ) ) ).

% ATP.lambda_386
tff(fact_10562_ATP_Olambda__387,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_abq(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub)),Uu)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_387
tff(fact_10563_ATP_Olambda__388,axiom,
    ! [Uu: set_Code_integer,Uua: nat,Uub: list_Code_integer] :
      ( aa_lis2813719629158071341eger_o(aa_nat2697030861539126eger_o(aTP_Lamm_abt(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_Code_integer2(Uub)),Uu)
        & ( size_s3445333598471063425nteger(Uub) = Uua ) ) ) ).

% ATP.lambda_388
tff(fact_10564_ATP_Olambda__389,axiom,
    ! [Uu: set_Pr5949110396991348497n_assn,Uua: nat,Uub: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(aa_nat2893684320935325170assn_o(aTP_Lamm_abr(Uu),Uua),Uub)
    <=> ( ord_le171416862856029873n_assn(set_Pr1139785259514867910n_assn(Uub),Uu)
        & ( size_s6829681357464350627n_assn(Uub) = Uua ) ) ) ).

% ATP.lambda_389
tff(fact_10565_ATP_Olambda__390,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_abs(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_390
tff(fact_10566_ATP_Olambda__391,axiom,
    ! [Uu: set_real,Uua: nat,Uub: list_real] :
      ( aa_list_real_o(aa_nat3629845878562903719real_o(aTP_Lamm_abu(Uu),Uua),Uub)
    <=> ( aa_set_real_o(ord_less_eq_set_real(set_real2(Uub)),Uu)
        & ( size_size_list_real(Uub) = Uua ) ) ) ).

% ATP.lambda_391
tff(fact_10567_ATP_Olambda__392,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_abv(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_o(set_o2(Uub),Uu)
        & ( size_size_list_o(Uub) = Uua ) ) ) ).

% ATP.lambda_392
tff(fact_10568_ATP_Olambda__393,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_abx(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu)
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_393
tff(fact_10569_ATP_Olambda__394,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_abw(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_394
tff(fact_10570_ATP_Olambda__395,axiom,
    ! [Uu: list_VEBT_VEBT,Uua: array_VEBT_VEBTi,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_arr7909941341466502432i_assn(aTP_Lamm_aqo(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,snga_assn_VEBT_VEBTi(Uua,Uub)),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uu,Uub)) ).

% ATP.lambda_395
tff(fact_10571_ATP_Olambda__396,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aov(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub)) ).

% ATP.lambda_396
tff(fact_10572_ATP_Olambda__397,axiom,
    ! [Uu: vEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_set2497645856053385599VEBT_o(aTP_Lamm_kw(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uua) ) ) ).

% ATP.lambda_397
tff(fact_10573_ATP_Olambda__398,axiom,
    ! [Uu: product_prod_int_int,Uua: set_Pr958786334691620121nt_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_set2224860389584585407_int_o(aTP_Lamm_lb(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_398
tff(fact_10574_ATP_Olambda__399,axiom,
    ! [Uu: complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_la(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_399
tff(fact_10575_ATP_Olambda__400,axiom,
    ! [Uu: real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_kx(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_400
tff(fact_10576_ATP_Olambda__401,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ky(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_401
tff(fact_10577_ATP_Olambda__402,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_kz(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_402
tff(fact_10578_ATP_Olambda__403,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_apd(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uua)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_403
tff(fact_10579_ATP_Olambda__404,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_acx(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_404
tff(fact_10580_ATP_Olambda__405,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ape(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_405
tff(fact_10581_ATP_Olambda__406,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_adb(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_406
tff(fact_10582_ATP_Olambda__407,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ada(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_407
tff(fact_10583_ATP_Olambda__408,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_acy(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_408
tff(fact_10584_ATP_Olambda__409,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_jf(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Uu),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uub) ) ) ).

% ATP.lambda_409
tff(fact_10585_ATP_Olambda__410,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_je(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Uu),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_410
tff(fact_10586_ATP_Olambda__411,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ari(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uua)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uu) ) ) ).

% ATP.lambda_411
tff(fact_10587_ATP_Olambda__412,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_arj(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uub),Uua)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uub),Uu) ) ) ).

% ATP.lambda_412
tff(fact_10588_ATP_Olambda__413,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer,Uub: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aa_set4044230130517860621eger_o(aTP_Lamm_abm(Uu),Uua),Uub)
    <=> ( minus_2355218937544613996nteger(Uub,Uu) = minus_2355218937544613996nteger(Uua,Uu) ) ) ).

% ATP.lambda_413
tff(fact_10589_ATP_Olambda__414,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: set_complex] :
      ( aa_set_complex_o(aa_set8643480048548153067plex_o(aTP_Lamm_abl(Uu),Uua),Uub)
    <=> ( minus_811609699411566653omplex(Uub,Uu) = minus_811609699411566653omplex(Uua,Uu) ) ) ).

% ATP.lambda_414
tff(fact_10590_ATP_Olambda__415,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_abn(Uu),Uua),Uub)
    <=> ( minus_minus_set_nat(Uub,Uu) = minus_minus_set_nat(Uua,Uu) ) ) ).

% ATP.lambda_415
tff(fact_10591_ATP_Olambda__416,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_abk(Uu),Uua),Uub)
    <=> ( minus_minus_set_int(Uub,Uu) = minus_minus_set_int(Uua,Uu) ) ) ).

% ATP.lambda_416
tff(fact_10592_ATP_Olambda__417,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_alq(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_417
tff(fact_10593_ATP_Olambda__418,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_aly(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & aa_Code_integer_o(Uua,Uub) ) ) ).

% ATP.lambda_418
tff(fact_10594_ATP_Olambda__419,axiom,
    ! [Uu: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_alw(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_419
tff(fact_10595_ATP_Olambda__420,axiom,
    ! [Uu: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_als(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_420
tff(fact_10596_ATP_Olambda__421,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_amc(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_421
tff(fact_10597_ATP_Olambda__422,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_alu(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_422
tff(fact_10598_ATP_Olambda__423,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_li(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_423
tff(fact_10599_ATP_Olambda__424,axiom,
    ! [Uu: product_prod_int_int,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_lk(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_424
tff(fact_10600_ATP_Olambda__425,axiom,
    ! [Uu: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_lj(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_425
tff(fact_10601_ATP_Olambda__426,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_ll(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_426
tff(fact_10602_ATP_Olambda__427,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_lm(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_427
tff(fact_10603_ATP_Olambda__428,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ln(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_428
tff(fact_10604_ATP_Olambda__429,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_lo(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_429
tff(fact_10605_ATP_Olambda__430,axiom,
    ! [Uu: product_prod_int_int,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_lq(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_430
tff(fact_10606_ATP_Olambda__431,axiom,
    ! [Uu: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_lp(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_431
tff(fact_10607_ATP_Olambda__432,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_lr(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_432
tff(fact_10608_ATP_Olambda__433,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ls(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_433
tff(fact_10609_ATP_Olambda__434,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_lt(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_434
tff(fact_10610_ATP_Olambda__435,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun1965331558590768990VEBT_o(aTP_Lamm_afa(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & ( aa_VEBT_VEBT_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_435
tff(fact_10611_ATP_Olambda__436,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun5889122065779655690VEBT_o(aTP_Lamm_afm(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & ( aa_VEBT_VEBT_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_436
tff(fact_10612_ATP_Olambda__437,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3960896333048504454r_real,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun3640226575523069072eger_o(aTP_Lamm_afk(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ( aa_Code_integer_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_437
tff(fact_10613_ATP_Olambda__438,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_afi(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_438
tff(fact_10614_ATP_Olambda__439,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_afc(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_439
tff(fact_10615_ATP_Olambda__440,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_afo(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_440
tff(fact_10616_ATP_Olambda__441,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_afe(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_441
tff(fact_10617_ATP_Olambda__442,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_afq(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_442
tff(fact_10618_ATP_Olambda__443,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_afg(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_443
tff(fact_10619_ATP_Olambda__444,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aTP_Lamm_afs(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_444
tff(fact_10620_ATP_Olambda__445,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_uint32,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun6272599145770227614VEBT_o(aTP_Lamm_aeg(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & ( aa_VEBT_VEBT_uint32(Uua,Uub) != one_one_uint32 ) ) ) ).

% ATP.lambda_445
tff(fact_10621_ATP_Olambda__446,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun1965331558590768990VEBT_o(aTP_Lamm_aes(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & ( aa_VEBT_VEBT_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_446
tff(fact_10622_ATP_Olambda__447,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co1554742298434871494uint32,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8926496376162642256eger_o(aTP_Lamm_aeq(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uub),Uu)
        & ( aa_Cod8401701657498852871uint32(Uua,Uub) != one_one_uint32 ) ) ) ).

% ATP.lambda_447
tff(fact_10623_ATP_Olambda__448,axiom,
    ! [Uu: set_complex,Uua: fun_complex_uint32,Uub: complex] :
      ( aa_complex_o(aa_fun436548534304416686plex_o(aTP_Lamm_aeo(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_uint32(Uua,Uub) != one_one_uint32 ) ) ) ).

% ATP.lambda_448
tff(fact_10624_ATP_Olambda__449,axiom,
    ! [Uu: set_real,Uua: fun_real_uint32,Uub: real] :
      ( aa_real_o(aa_fun7556296611544133298real_o(aTP_Lamm_aei(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_uint32(Uua,Uub) != one_one_uint32 ) ) ) ).

% ATP.lambda_449
tff(fact_10625_ATP_Olambda__450,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_aeu(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_450
tff(fact_10626_ATP_Olambda__451,axiom,
    ! [Uu: set_nat,Uua: fun_nat_uint32,Uub: nat] :
      ( aa_nat_o(aa_fun1182053767699555178_nat_o(aTP_Lamm_aek(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_uint32(Uua,Uub) != one_one_uint32 ) ) ) ).

% ATP.lambda_451
tff(fact_10627_ATP_Olambda__452,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_aew(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_452
tff(fact_10628_ATP_Olambda__453,axiom,
    ! [Uu: set_int,Uua: fun_int_uint32,Uub: int] :
      ( aa_int_o(aa_fun6992502331602369202_int_o(aTP_Lamm_aem(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_uint32(Uua,Uub) != one_one_uint32 ) ) ) ).

% ATP.lambda_453
tff(fact_10629_ATP_Olambda__454,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_aey(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_454
tff(fact_10630_ATP_Olambda__455,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: $o] : aa_o_assn(aa_VEB6687300313434771872o_assn(aTP_Lamm_aqg(Uu),Uua),(Uub)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),pure_assn((Uub) = aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Uu))) ).

% ATP.lambda_455
tff(fact_10631_ATP_Olambda__456,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_aqe(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),pure_assn(Uub = vEBT_vebt_mint(Uu))) ).

% ATP.lambda_456
tff(fact_10632_ATP_Olambda__457,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_aqf(Uu),Uua),Uub) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),pure_assn(Uub = vEBT_vebt_maxt(Uu))) ).

% ATP.lambda_457
tff(fact_10633_ATP_Olambda__458,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_set2497645856053385599VEBT_o(aTP_Lamm_jo(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uua) ) ) ).

% ATP.lambda_458
tff(fact_10634_ATP_Olambda__459,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_set2224860389584585407_int_o(aTP_Lamm_js(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & ~ member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_459
tff(fact_10635_ATP_Olambda__460,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_jr(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ~ aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_460
tff(fact_10636_ATP_Olambda__461,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_jp(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ~ aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_461
tff(fact_10637_ATP_Olambda__462,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_jt(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ~ aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_462
tff(fact_10638_ATP_Olambda__463,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_jq(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ~ aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_463
tff(fact_10639_ATP_Olambda__464,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_arh(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uu) ) ).

% ATP.lambda_464
tff(fact_10640_ATP_Olambda__465,axiom,
    ! [Uu: set_Pr8218934625190621173um_num,Uua: num,Uub: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_sj(Uu),Uua),Uub)
    <=> member7279096912039735102um_num(aa_num3156681643174609981um_num(product_Pair_num_num(Uua),Uub),Uu) ) ).

% ATP.lambda_465
tff(fact_10641_ATP_Olambda__466,axiom,
    ! [Uu: set_Pr563407847431865468T_VEBT,Uua: nat,Uub: produc4813437837504472865T_VEBT] :
      ( aa_Pro3461009536670225267VEBT_o(aa_nat6863640065291604488VEBT_o(aTP_Lamm_sk(Uu),Uua),Uub)
    <=> member306291179834725981T_VEBT(aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(Uua),Uub),Uu) ) ).

% ATP.lambda_466
tff(fact_10642_ATP_Olambda__467,axiom,
    ! [Uu: set_Pr6200539531224447659at_num,Uua: nat,Uub: num] :
      ( aa_num_o(aa_nat_fun_num_o(aTP_Lamm_sl(Uu),Uua),Uub)
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(product_Pair_nat_num(Uua),Uub),Uu) ) ).

% ATP.lambda_467
tff(fact_10643_ATP_Olambda__468,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_sm(Uu),Uua),Uub)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub),Uu) ) ).

% ATP.lambda_468
tff(fact_10644_ATP_Olambda__469,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_sn(Uu),Uua),Uub)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub),Uu) ) ).

% ATP.lambda_469
tff(fact_10645_ATP_Olambda__470,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_acz(Uu),Uua),Uub)
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uu) = Uua ) ) ).

% ATP.lambda_470
tff(fact_10646_ATP_Olambda__471,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_api(Uu),Uua),Uub)
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uua) = Uu ) ) ).

% ATP.lambda_471
tff(fact_10647_ATP_Olambda__472,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_vv(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,aa_nat_nat(suc,Uub))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub)) ).

% ATP.lambda_472
tff(fact_10648_ATP_Olambda__473,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vw(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,aa_nat_nat(suc,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_473
tff(fact_10649_ATP_Olambda__474,axiom,
    ! [Uu: fun_fu7983188961739674408Heap_o,Uua: fun_o_2166024663187074044_VEBTi,Uub: fun_Pr342922516890243977on_nat] : aa_fun765629416642892858_VEBTi(aa_fun565820801828592151_VEBTi(aTP_Lamm_rd(Uu),Uua),Uub) = heap_T5998771940306268294_VEBTi(aa_fun4188577930510380905Heap_o(Uu,Uub),aa_fun8426004761281959347_VEBTi(aTP_Lamm_rc(Uua),Uub)) ).

% ATP.lambda_474
tff(fact_10650_ATP_Olambda__475,axiom,
    ! [Uu: fun_fu7983188961739674408Heap_o,Uua: fun_o_5993975905899854163Heap_o,Uub: fun_Pr342922516890243977on_nat] : aa_fun4188577930510380905Heap_o(aa_fun8877585628723769663Heap_o(aTP_Lamm_rb(Uu),Uua),Uub) = heap_Time_bind_o_o(aa_fun4188577930510380905Heap_o(Uu,Uub),aa_fun7300956949533034054Heap_o(aTP_Lamm_ra(Uua),Uub)) ).

% ATP.lambda_475
tff(fact_10651_ATP_Olambda__476,axiom,
    ! [Uu: fun_fu2606674006500944492ap_nat,Uua: fun_na7267038673063590618_VEBTi,Uub: fun_Pr342922516890243977on_nat] : aa_fun765629416642892858_VEBTi(aa_fun3907257510474286387_VEBTi(aTP_Lamm_qz(Uu),Uua),Uub) = heap_T844888390831797134_VEBTi(aa_fun4282859208648741077ap_nat(Uu,Uub),aa_fun1166700169754672579_VEBTi(aTP_Lamm_qy(Uua),Uub)) ).

% ATP.lambda_476
tff(fact_10652_ATP_Olambda__477,axiom,
    ! [Uu: fun_fu2606674006500944492ap_nat,Uua: fun_na979242554834862325Heap_o,Uub: fun_Pr342922516890243977on_nat] : aa_fun4188577930510380905Heap_o(aa_fun7768191950043460187Heap_o(aTP_Lamm_qx(Uu),Uua),Uub) = heap_Time_bind_nat_o(aa_fun4282859208648741077ap_nat(Uu,Uub),aa_fun1690681901636463158Heap_o(aTP_Lamm_qw(Uua),Uub)) ).

% ATP.lambda_477
tff(fact_10653_ATP_Olambda__478,axiom,
    ! [Uu: fun_fu335202653675969872Heap_o,Uua: fun_o_2223814739247024763Heap_o,Uub: fun_Pr238758252084143515Heap_o] : aa_fun6848502500920544641Heap_o(aa_fun4354595424708498391Heap_o(aTP_Lamm_qv(Uu),Uua),Uub) = heap_Time_bind_o_o(aa_fun6848502500920544641Heap_o(Uu,Uub),aa_fun7917812164634591790Heap_o(aTP_Lamm_qu(Uua),Uub)) ).

% ATP.lambda_478
tff(fact_10654_ATP_Olambda__479,axiom,
    ! [Uu: fun_fu335202653675969872Heap_o,Uua: fun_o_7171189317104958676_VEBTi,Uub: fun_Pr238758252084143515Heap_o] : aa_fun238413853627933730_VEBTi(aa_fun2042045670669332929_VEBTi(aTP_Lamm_qt(Uu),Uua),Uub) = heap_T5998771940306268294_VEBTi(aa_fun6848502500920544641Heap_o(Uu,Uub),aa_fun6586487760926256075_VEBTi(aTP_Lamm_qs(Uua),Uub)) ).

% ATP.lambda_479
tff(fact_10655_ATP_Olambda__480,axiom,
    ! [Uu: fun_fu4424605311360796457_VEBTi,Uua: fun_VE6943415934277719092Heap_o,Uub: fun_Pr238758252084143515Heap_o] : aa_fun6848502500920544641Heap_o(aa_fun5382224708498380120Heap_o(aTP_Lamm_qr(Uu),Uua),Uub) = heap_T3040810144269856602EBTi_o(aa_fun238413853627933730_VEBTi(Uu,Uub),aa_fun2163459272138508211Heap_o(aTP_Lamm_qq(Uua),Uub)) ).

% ATP.lambda_480
tff(fact_10656_ATP_Olambda__481,axiom,
    ! [Uu: fun_fu5786974620624140868ap_nat,Uua: fun_na8517577583218006873Heap_o,Uub: fun_Pr238758252084143515Heap_o] : aa_fun6848502500920544641Heap_o(aa_fun3258944081180477171Heap_o(aTP_Lamm_qp(Uu),Uua),Uub) = heap_Time_bind_nat_o(aa_fun6684961993577261245ap_nat(Uu,Uub),aa_fun7995322860690487374Heap_o(aTP_Lamm_qo(Uua),Uub)) ).

% ATP.lambda_481
tff(fact_10657_ATP_Olambda__482,axiom,
    ! [Uu: fun_fu4424605311360796457_VEBTi,Uua: fun_VE6491116402277797979_VEBTi,Uub: fun_Pr238758252084143515Heap_o] : aa_fun238413853627933730_VEBTi(aa_fun4276395798348166338_VEBTi(aTP_Lamm_qn(Uu),Uua),Uub) = heap_T1006145433769338483_VEBTi(aa_fun238413853627933730_VEBTi(Uu,Uub),aa_fun8469889001190476422_VEBTi(aTP_Lamm_qm(Uua),Uub)) ).

% ATP.lambda_482
tff(fact_10658_ATP_Olambda__483,axiom,
    ! [Uu: fun_fu5786974620624140868ap_nat,Uua: fun_na5845961453008041334_VEBTi,Uub: fun_Pr238758252084143515Heap_o] : aa_fun238413853627933730_VEBTi(aa_fun90388479857626845_VEBTi(aTP_Lamm_ql(Uu),Uua),Uub) = heap_T844888390831797134_VEBTi(aa_fun6684961993577261245ap_nat(Uu,Uub),aa_fun5587972357748661675_VEBTi(aTP_Lamm_qk(Uua),Uub)) ).

% ATP.lambda_483
tff(fact_10659_ATP_Olambda__484,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_atu(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(times_times_real(numeral_numeral_real(aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_484
tff(fact_10660_ATP_Olambda__485,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ate(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_485
tff(fact_10661_ATP_Olambda__486,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_vs(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub)) ).

% ATP.lambda_486
tff(fact_10662_ATP_Olambda__487,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vq(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_487
tff(fact_10663_ATP_Olambda__488,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_apa(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ).

% ATP.lambda_488
tff(fact_10664_ATP_Olambda__489,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_abo(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_489
tff(fact_10665_ATP_Olambda__490,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aua(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_490
tff(fact_10666_ATP_Olambda__491,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_aud(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_491
tff(fact_10667_ATP_Olambda__492,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aqp(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_492
tff(fact_10668_ATP_Olambda__493,axiom,
    ! [Uu: fun_VEBT_VEBT_int,Uua: fun_VEBT_VEBT_int,Uub: vEBT_VEBT] : aa_VEBT_VEBT_int(aa_fun7144222995531728252BT_int(aTP_Lamm_any(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_VEBT_VEBT_int(Uua,Uub)),aa_VEBT_VEBT_int(Uu,Uub)) ).

% ATP.lambda_493
tff(fact_10669_ATP_Olambda__494,axiom,
    ! [Uu: fun_real_int,Uua: fun_real_int,Uub: real] : aa_real_int(aa_fun8270430325945277160al_int(aTP_Lamm_anz(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(Uua,Uub)),aa_real_int(Uu,Uub)) ).

% ATP.lambda_494
tff(fact_10670_ATP_Olambda__495,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_aoa(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_495
tff(fact_10671_ATP_Olambda__496,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_aob(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uua,Uub)),aa_int_int(Uu,Uub)) ).

% ATP.lambda_496
tff(fact_10672_ATP_Olambda__497,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_aix(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_497
tff(fact_10673_ATP_Olambda__498,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vj(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(minus_minus_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_498
tff(fact_10674_ATP_Olambda__499,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vk(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_499
tff(fact_10675_ATP_Olambda__500,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_aiy(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_500
tff(fact_10676_ATP_Olambda__501,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_aih(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_501
tff(fact_10677_ATP_Olambda__502,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_vb(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(plus_plus_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_502
tff(fact_10678_ATP_Olambda__503,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vc(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_503
tff(fact_10679_ATP_Olambda__504,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_vd(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_504
tff(fact_10680_ATP_Olambda__505,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ve(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_505
tff(fact_10681_ATP_Olambda__506,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_aii(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_506
tff(fact_10682_ATP_Olambda__507,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ast(Uu),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_507
tff(fact_10683_ATP_Olambda__508,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_acr(Uu),Uua),Uub)
    <=> ( aa_Code_integer_o(Uu,Uub)
        | aa_Code_integer_o(Uua,Uub) ) ) ).

% ATP.lambda_508
tff(fact_10684_ATP_Olambda__509,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_acn(Uu),Uua),Uub)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        | aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_509
tff(fact_10685_ATP_Olambda__510,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_acq(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        | aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_510
tff(fact_10686_ATP_Olambda__511,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_aco(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        | aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_511
tff(fact_10687_ATP_Olambda__512,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_acp(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        | aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_512
tff(fact_10688_ATP_Olambda__513,axiom,
    ! [Uu: fun_Code_integer_o,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_acw(Uu),Uua),Uub)
    <=> ( aa_Code_integer_o(Uu,Uub)
        & aa_Code_integer_o(Uua,Uub) ) ) ).

% ATP.lambda_513
tff(fact_10689_ATP_Olambda__514,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_acs(Uu),Uua),Uub)
    <=> ( aa_Pro4592068669816069595_int_o(Uu,Uub)
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_514
tff(fact_10690_ATP_Olambda__515,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_acv(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_515
tff(fact_10691_ATP_Olambda__516,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_act(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_516
tff(fact_10692_ATP_Olambda__517,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_acu(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_517
tff(fact_10693_ATP_Olambda__518,axiom,
    ! [Uu: fun_co7673397065115134084omplex,Uua: set_complex,Uub: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aTP_Lamm_ahx(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(Uu,Uub),Uua) ).

% ATP.lambda_518
tff(fact_10694_ATP_Olambda__519,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: set_nat,Uub: nat] : aa_nat_real(aa_set1166323925324055613t_real(aTP_Lamm_aia(Uu),Uua),Uub) = groups6591440286371151544t_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_519
tff(fact_10695_ATP_Olambda__520,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: set_nat,Uub: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aTP_Lamm_ahu(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_520
tff(fact_10696_ATP_Olambda__521,axiom,
    ! [Uu: fun_int_fun_int_int,Uua: set_int,Uub: int] : aa_int_int(aa_set7310040872927613301nt_int(aTP_Lamm_aid(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_int_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_521
tff(fact_10697_ATP_Olambda__522,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_abp(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_522
tff(fact_10698_ATP_Olambda__523,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_akw(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_523
tff(fact_10699_ATP_Olambda__524,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_akx(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_524
tff(fact_10700_ATP_Olambda__525,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_aiz(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_525
tff(fact_10701_ATP_Olambda__526,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_uv(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(divide1717551699836669952omplex,aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_526
tff(fact_10702_ATP_Olambda__527,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uw(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_527
tff(fact_10703_ATP_Olambda__528,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_ais(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_528
tff(fact_10704_ATP_Olambda__529,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_vf(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_529
tff(fact_10705_ATP_Olambda__530,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vg(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_530
tff(fact_10706_ATP_Olambda__531,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_air(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_531
tff(fact_10707_ATP_Olambda__532,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ait(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_532
tff(fact_10708_ATP_Olambda__533,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_asq(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_533
tff(fact_10709_ATP_Olambda__534,axiom,
    ! [Uu: fun_li7499208429157053461_VEBTi,Uua: fun_VE4765431424076933413_VEBTi,Uub: list_VEBT_VEBTi] : aa_lis1839255523752068350_VEBTi(aa_fun2525796888956276588_VEBTi(aTP_Lamm_qj(Uu),Uua),Uub) = heap_T1006145433769338483_VEBTi(aa_lis1839255523752068350_VEBTi(Uu,Uub),Uua) ).

% ATP.lambda_534
tff(fact_10710_ATP_Olambda__535,axiom,
    ! [Uu: fun_o_2238389867833408620_VEBTi,Uua: fun_VE4765431424076933413_VEBTi,Uub: $o] : aa_o_H7577119287286566245_VEBTi(aa_fun432179723158315733_VEBTi(aTP_Lamm_qa(Uu),Uua),(Uub)) = heap_T1006145433769338483_VEBTi(aa_o_H7577119287286566245_VEBTi(Uu,(Uub)),Uua) ).

% ATP.lambda_535
tff(fact_10711_ATP_Olambda__536,axiom,
    ! [Uu: fun_o_8325599348076409421Heap_o,Uua: fun_o_2238389867833408620_VEBTi,Uub: $o] : aa_o_H7577119287286566245_VEBTi(aa_fun5660287117450118806_VEBTi(aTP_Lamm_qe(Uu),Uua),(Uub)) = heap_T5998771940306268294_VEBTi(aa_o_H7399033255965945470Heap_o(Uu,(Uub)),Uua) ).

% ATP.lambda_536
tff(fact_10712_ATP_Olambda__537,axiom,
    ! [Uu: fun_o_4158591058828729991ap_nat,Uua: fun_na7848093160509028170_VEBTi,Uub: $o] : aa_o_H7577119287286566245_VEBTi(aa_fun1800422248244663418_VEBTi(aTP_Lamm_qg(Uu),Uua),(Uub)) = heap_T844888390831797134_VEBTi(aa_o_H8739796202006397184ap_nat(Uu,(Uub)),Uua) ).

% ATP.lambda_537
tff(fact_10713_ATP_Olambda__538,axiom,
    ! [Uu: fun_na7848093160509028170_VEBTi,Uua: fun_VE4765431424076933413_VEBTi,Uub: nat] : aa_nat5408703056993926195_VEBTi(aa_fun6311797650338102561_VEBTi(aTP_Lamm_qb(Uu),Uua),Uub) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(Uu,Uub),Uua) ).

% ATP.lambda_538
tff(fact_10714_ATP_Olambda__539,axiom,
    ! [Uu: fun_na7848093160509028170_VEBTi,Uua: fun_VE7038733415900436564Heap_o,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_fun3725866701989745153Heap_o(aTP_Lamm_qc(Uu),Uua),Uub) = heap_T3040810144269856602EBTi_o(aa_nat5408703056993926195_VEBTi(Uu,Uub),Uua) ).

% ATP.lambda_539
tff(fact_10715_ATP_Olambda__540,axiom,
    ! [Uu: fun_na2343073368905879023Heap_o,Uua: fun_o_2238389867833408620_VEBTi,Uub: nat] : aa_nat5408703056993926195_VEBTi(aa_fun5487288040762657696_VEBTi(aTP_Lamm_qf(Uu),Uua),Uub) = heap_T5998771940306268294_VEBTi(aa_nat2910048432523430448Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_540
tff(fact_10716_ATP_Olambda__541,axiom,
    ! [Uu: fun_na2343073368905879023Heap_o,Uua: fun_o_8325599348076409421Heap_o,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_fun1108420279618462208Heap_o(aTP_Lamm_qd(Uu),Uua),Uub) = heap_Time_bind_o_o(aa_nat2910048432523430448Heap_o(Uu,Uub),Uua) ).

% ATP.lambda_541
tff(fact_10717_ATP_Olambda__542,axiom,
    ! [Uu: fun_na3533371890617757541ap_nat,Uua: fun_na7848093160509028170_VEBTi,Uub: nat] : aa_nat5408703056993926195_VEBTi(aa_fun49764924992962236_VEBTi(aTP_Lamm_qh(Uu),Uua),Uub) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(Uu,Uub),Uua) ).

% ATP.lambda_542
tff(fact_10718_ATP_Olambda__543,axiom,
    ! [Uu: fun_na3533371890617757541ap_nat,Uua: fun_na2343073368905879023Heap_o,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_fun3424352875297450780Heap_o(aTP_Lamm_qi(Uu),Uua),Uub) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(Uu,Uub),Uua) ).

% ATP.lambda_543
tff(fact_10719_ATP_Olambda__544,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ass(Uu),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_544
tff(fact_10720_ATP_Olambda__545,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_VEBT_VEBTi] : aa_lis1839255523752068350_VEBTi(aa_nat4125268785154830727_VEBTi(aTP_Lamm_aax(Uu),Uua),Uub) = heap_T7982501707604696571_VEBTi(refine_Imp_assert(size_s7982070591426661849_VEBTi(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,Uua))),aa_lis4026867407995182650_VEBTi(aa_nat1561540599483812487_VEBTi(aTP_Lamm_aaw(Uu),Uua),Uub)) ).

% ATP.lambda_545
tff(fact_10721_ATP_Olambda__546,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_VEBT_VEBTi] : aa_lis1839255523752068350_VEBTi(aa_nat4125268785154830727_VEBTi(aTP_Lamm_aau(Uu),Uua),Uub) = heap_T7982501707604696571_VEBTi(refine_Imp_assert(size_s7982070591426661849_VEBTi(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),aa_lis4026867407995182650_VEBTi(aa_nat1561540599483812487_VEBTi(aTP_Lamm_aat(Uu),Uua),Uub)) ).

% ATP.lambda_546
tff(fact_10722_ATP_Olambda__547,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_lc(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_547
tff(fact_10723_ATP_Olambda__548,axiom,
    ! [Uu: product_prod_int_int,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_lh(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_548
tff(fact_10724_ATP_Olambda__549,axiom,
    ! [Uu: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_lg(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_549
tff(fact_10725_ATP_Olambda__550,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_ld(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_550
tff(fact_10726_ATP_Olambda__551,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_le(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_551
tff(fact_10727_ATP_Olambda__552,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_lf(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_552
tff(fact_10728_ATP_Olambda__553,axiom,
    ! [Uu: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_vz(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_553
tff(fact_10729_ATP_Olambda__554,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi] : aa_arr4839786837114051083_VEBTi(aa_nat106358831840921632_VEBTi(aTP_Lamm_aav(Uu),Uua),Uub) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,aa_nat_nat(suc,Uua)),aa_arr2912372873376718191_VEBTi(aTP_Lamm_aar(Uu),Uub)) ).

% ATP.lambda_554
tff(fact_10730_ATP_Olambda__555,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi] : aa_arr4839786837114051083_VEBTi(aa_nat106358831840921632_VEBTi(aTP_Lamm_aba(Uu),Uua),Uub) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,aa_nat_nat(suc,Uua)),aa_arr2912372873376718191_VEBTi(aTP_Lamm_aar(Uu),Uub)) ).

% ATP.lambda_555
tff(fact_10731_ATP_Olambda__556,axiom,
    ! [Uu: array_VEBT_VEBTi,Uua: nat,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aTP_Lamm_nu(Uu),Uua),Uub) = heap_Time_bind_nat_o(array_len_VEBT_VEBTi(Uu),aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aTP_Lamm_nt(Uu),Uua),Uub)) ).

% ATP.lambda_556
tff(fact_10732_ATP_Olambda__557,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi] : aa_arr4839786837114051083_VEBTi(aa_nat106358831840921632_VEBTi(aTP_Lamm_aas(Uu),Uua),Uub) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(vEBT_V739175172307565963ildupi,Uua),aa_arr2912372873376718191_VEBTi(aTP_Lamm_aar(Uu),Uub)) ).

% ATP.lambda_557
tff(fact_10733_ATP_Olambda__558,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi] : aa_arr4839786837114051083_VEBTi(aa_nat106358831840921632_VEBTi(aTP_Lamm_aay(Uu),Uua),Uub) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(vEBT_vebt_buildupi,Uua),aa_arr2912372873376718191_VEBTi(aTP_Lamm_aar(Uu),Uub)) ).

% ATP.lambda_558
tff(fact_10734_ATP_Olambda__559,axiom,
    ! [Uu: nat,Uua: option_nat,Uub: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_opt1214731524852023858on_nat(aTP_Lamm_ci(Uu),Uua),Uub) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uub),aa_opt7201247987542299399on_nat(aTP_Lamm_aa(Uu),Uua)) ).

% ATP.lambda_559
tff(fact_10735_ATP_Olambda__560,axiom,
    ! [Uu: nat,Uua: option_nat,Uub: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_opt1214731524852023858on_nat(aTP_Lamm_ab(Uu),Uua),Uub) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Uub),aa_opt7201247987542299399on_nat(aTP_Lamm_aa(Uu),Uua)) ).

% ATP.lambda_560
tff(fact_10736_ATP_Olambda__561,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_VEBT_VEBTi] : aa_lis1839255523752068350_VEBTi(aa_nat4125268785154830727_VEBTi(aTP_Lamm_abb(Uu),Uua),Uub) = heap_T5099337393651448672_VEBTi(array_615059503499738533_VEBTi(Uub),aa_nat106358831840921632_VEBTi(aTP_Lamm_aba(Uu),Uua)) ).

% ATP.lambda_561
tff(fact_10737_ATP_Olambda__562,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_VEBT_VEBTi] : aa_lis1839255523752068350_VEBTi(aa_nat4125268785154830727_VEBTi(aTP_Lamm_aaz(Uu),Uua),Uub) = heap_T5099337393651448672_VEBTi(array_615059503499738533_VEBTi(Uub),aa_nat106358831840921632_VEBTi(aTP_Lamm_aay(Uu),Uua)) ).

% ATP.lambda_562
tff(fact_10738_ATP_Olambda__563,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ato(Uu),Uua),Uub) = aa_real_real(divide_divide_real(Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_563
tff(fact_10739_ATP_Olambda__564,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_and(Uu),Uua),Uub) = aa_nat_Code_integer(power_8256067586552552935nteger(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_564
tff(fact_10740_ATP_Olambda__565,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_anc(Uu),Uua),Uub) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_565
tff(fact_10741_ATP_Olambda__566,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ang(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_566
tff(fact_10742_ATP_Olambda__567,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_ane(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_567
tff(fact_10743_ATP_Olambda__568,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_anf(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_568
tff(fact_10744_ATP_Olambda__569,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aoc(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_569
tff(fact_10745_ATP_Olambda__570,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_aub(Uu),Uua),Uub)
    <=> aa_set_real_o(member_real(Uub),set_or1633881224788618240n_real(Uu,Uua)) ) ).

% ATP.lambda_570
tff(fact_10746_ATP_Olambda__571,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_aiv(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Uu),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_571
tff(fact_10747_ATP_Olambda__572,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ut(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_572
tff(fact_10748_ATP_Olambda__573,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_uu(Uu),Uua),Uub) = aa_real_real(times_times_real(Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_573
tff(fact_10749_ATP_Olambda__574,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_aiu(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_574
tff(fact_10750_ATP_Olambda__575,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_aiw(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,Uu),aa_int_int(Uua,Uub)) ).

% ATP.lambda_575
tff(fact_10751_ATP_Olambda__576,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_vh(Uu),Uua),Uub) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,Uua),aa_nat_complex(Uu,Uub)) ).

% ATP.lambda_576
tff(fact_10752_ATP_Olambda__577,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vi(Uu),Uua),Uub) = aa_real_real(times_times_real(Uua),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_577
tff(fact_10753_ATP_Olambda__578,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer,Uub: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_aea(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_Code_integer_nat(Uu,Uub)),aa_Code_integer_nat(Uu,Uub))) ) ).

% ATP.lambda_578
tff(fact_10754_ATP_Olambda__579,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: product_prod_int_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(aTP_Lamm_adw(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_Pro6801748904974264397nt_nat(Uu,Uub)),aa_Pro6801748904974264397nt_nat(Uu,Uub))) ) ).

% ATP.lambda_579
tff(fact_10755_ATP_Olambda__580,axiom,
    ! [Uu: fun_complex_nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_adz(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_complex_nat(Uu,Uub)),aa_complex_nat(Uu,Uub))) ) ).

% ATP.lambda_580
tff(fact_10756_ATP_Olambda__581,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_adx(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uu,Uub))) ) ).

% ATP.lambda_581
tff(fact_10757_ATP_Olambda__582,axiom,
    ! [Uu: fun_int_nat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ady(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_int_nat(Uu,Uub)),aa_int_nat(Uu,Uub))) ) ).

% ATP.lambda_582
tff(fact_10758_ATP_Olambda__583,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_aef(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_Code_integer_o(Uua,Uub),aa_Code_integer_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_583
tff(fact_10759_ATP_Olambda__584,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_aeb(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_Pro4592068669816069595_int_o(Uua,Uub),aa_Pro6801748904974264397nt_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_584
tff(fact_10760_ATP_Olambda__585,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_aee(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_complex_o(Uua,Uub),aa_complex_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_585
tff(fact_10761_ATP_Olambda__586,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_aec(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_nat_o(Uua,Uub),aa_nat_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_586
tff(fact_10762_ATP_Olambda__587,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_aed(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_int_o(Uua,Uub),aa_int_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_587
tff(fact_10763_ATP_Olambda__588,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_nat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun4054834855839773292eger_o(aTP_Lamm_adv(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Code_integer_nat(Uu,Uub)),aa_Code_integer_nat(Uua,Uub))) ) ).

% ATP.lambda_588
tff(fact_10764_ATP_Olambda__589,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: fun_Pr5160692531401422620nt_nat,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun6302685600795359042_int_o(aTP_Lamm_adn(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Pro6801748904974264397nt_nat(Uu,Uub)),aa_Pro6801748904974264397nt_nat(Uua,Uub))) ) ).

% ATP.lambda_589
tff(fact_10765_ATP_Olambda__590,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] :
      ( aa_complex_o(aa_fun8723867720292580170plex_o(aTP_Lamm_adt(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uu,Uub)),aa_complex_nat(Uua,Uub))) ) ).

% ATP.lambda_590
tff(fact_10766_ATP_Olambda__591,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_adp(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub))) ) ).

% ATP.lambda_591
tff(fact_10767_ATP_Olambda__592,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_adr(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uu,Uub)),aa_int_nat(Uua,Uub))) ) ).

% ATP.lambda_592
tff(fact_10768_ATP_Olambda__593,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_us(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_593
tff(fact_10769_ATP_Olambda__594,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ur(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_594
tff(fact_10770_ATP_Olambda__595,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aty(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_595
tff(fact_10771_ATP_Olambda__596,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_amh(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_596
tff(fact_10772_ATP_Olambda__597,axiom,
    ! [Uu: fun_Pr2118491707810114563at_nat,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_dl(Uu),Uua),Uub) = aa_Pro262675715343573244at_nat(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ).

% ATP.lambda_597
tff(fact_10773_ATP_Olambda__598,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_dm(Uu),Uua),Uub)
    <=> aa_Pro6150637604445964947_nat_o(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ) ).

% ATP.lambda_598
tff(fact_10774_ATP_Olambda__599,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dn(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_599
tff(fact_10775_ATP_Olambda__600,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_do(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ) ).

% ATP.lambda_600
tff(fact_10776_ATP_Olambda__601,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_dp(Uu),Uua),Uub) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_601
tff(fact_10777_ATP_Olambda__602,axiom,
    ! [Uu: fun_op8989173174992673105on_num,Uua: fun_num_option_num,Uub: num] : aa_num_option_num(aa_fun3636977833914307392on_num(aTP_Lamm_dj(Uu),Uua),Uub) = aa_opt5369159329878751634on_num(Uu,aa_num_option_num(Uua,Uub)) ).

% ATP.lambda_602
tff(fact_10778_ATP_Olambda__603,axiom,
    ! [Uu: fun_option_num_num,Uua: fun_num_option_num,Uub: num] : aa_num_num(aa_fun7142274290070961008um_num(aTP_Lamm_df(Uu),Uua),Uub) = aa_option_num_num(Uu,aa_num_option_num(Uua,Uub)) ).

% ATP.lambda_603
tff(fact_10779_ATP_Olambda__604,axiom,
    ! [Uu: fun_option_num_int,Uua: fun_num_option_num,Uub: num] : aa_num_int(aa_fun2256179333612926594um_int(aTP_Lamm_dg(Uu),Uua),Uub) = aa_option_num_int(Uu,aa_num_option_num(Uua,Uub)) ).

% ATP.lambda_604
tff(fact_10780_ATP_Olambda__605,axiom,
    ! [Uu: fun_o_o,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_dk(Uu),Uua),Uub)
    <=> aa_o_o(Uu,aa_Pro6150637604445964947_nat_o(Uua,Uub)) ) ).

% ATP.lambda_605
tff(fact_10781_ATP_Olambda__606,axiom,
    ! [Uu: fun_num_option_num,Uua: fun_num_num,Uub: num] : aa_num_option_num(aa_fun612943910911948912on_num(aTP_Lamm_dh(Uu),Uua),Uub) = aa_num_option_num(Uu,aa_num_num(Uua,Uub)) ).

% ATP.lambda_606
tff(fact_10782_ATP_Olambda__607,axiom,
    ! [Uu: fun_num_num,Uua: fun_num_num,Uub: num] : aa_num_num(aa_fun8048995283964520608um_num(aTP_Lamm_db(Uu),Uua),Uub) = aa_num_num(Uu,aa_num_num(Uua,Uub)) ).

% ATP.lambda_607
tff(fact_10783_ATP_Olambda__608,axiom,
    ! [Uu: fun_num_int,Uua: fun_num_num,Uub: num] : aa_num_int(aa_fun3162900327506486194um_int(aTP_Lamm_dc(Uu),Uua),Uub) = aa_num_int(Uu,aa_num_num(Uua,Uub)) ).

% ATP.lambda_608
tff(fact_10784_ATP_Olambda__609,axiom,
    ! [Uu: fun_int_option_num,Uua: fun_num_int,Uub: num] : aa_num_option_num(aa_fun3858197967482829954on_num(aTP_Lamm_di(Uu),Uua),Uub) = aa_int_option_num(Uu,aa_num_int(Uua,Uub)) ).

% ATP.lambda_609
tff(fact_10785_ATP_Olambda__610,axiom,
    ! [Uu: fun_int_num,Uua: fun_num_int,Uub: num] : aa_num_num(aa_fun5215869348392759218um_num(aTP_Lamm_dd(Uu),Uua),Uub) = aa_int_num(Uu,aa_num_int(Uua,Uub)) ).

% ATP.lambda_610
tff(fact_10786_ATP_Olambda__611,axiom,
    ! [Uu: fun_int_int,Uua: fun_num_int,Uub: num] : aa_num_int(aa_fun329774391934724804um_int(aTP_Lamm_de(Uu),Uua),Uub) = aa_int_int(Uu,aa_num_int(Uua,Uub)) ).

% ATP.lambda_611
tff(fact_10787_ATP_Olambda__612,axiom,
    ! [Uu: fun_na1476630412504255114at_nat,Uua: nat,Uub: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_hp(Uu),Uua),Uub) = aa_nat7603221163087113075at_nat(Uu,Uua) ).

% ATP.lambda_612
tff(fact_10788_ATP_Olambda__613,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_hr(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,Uua) ) ).

% ATP.lambda_613
tff(fact_10789_ATP_Olambda__614,axiom,
    ! [Uu: fun_in4211310926729683942nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ht(Uu),Uua),Uub) = aa_int7462565939943104719nt_int(Uu,Uua) ).

% ATP.lambda_614
tff(fact_10790_ATP_Olambda__615,axiom,
    ! [Uu: fun_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_hv(Uu),Uua),Uub)
    <=> aa_int_o(Uu,Uua) ) ).

% ATP.lambda_615
tff(fact_10791_ATP_Olambda__616,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_hx(Uu),Uua),Uub) = aa_int_int(Uu,Uua) ).

% ATP.lambda_616
tff(fact_10792_ATP_Olambda__617,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aTP_Lamm_aar(Uu),Uua),Uub) = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Uu)),Uua),Uub)) ).

% ATP.lambda_617
tff(fact_10793_ATP_Olambda__618,axiom,
    ! [Uu: nat,Uua: $o,Uub: $o] :
      aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(aTP_Lamm_nr(Uu),(Uua)),(Uub)) = aa_o_H7399033255965945470Heap_o(heap_Time_return_o,
        $ite(
          Uu = zero_zero_nat,
          (Uua),
          $ite(Uu = one_one_nat,(Uub),$false) )) ).

% ATP.lambda_618
tff(fact_10794_ATP_Olambda__619,axiom,
    ! [Uu: nat,Uua: nat,Uub: option_nat] : aa_opt9163293676379396302on_nat(aa_nat9127621225095157975on_nat(aTP_Lamm_a(Uu),Uua),Uub) = heap_T3487192422709364219on_nat(aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_option_nat(some_nat,Uua))),Uub)) ).

% ATP.lambda_619
tff(fact_10795_ATP_Olambda__620,axiom,
    ! [Uu: nat,Uua: option_nat,Uub: option_nat] : aa_opt9163293676379396302on_nat(aa_opt7201247987542299399on_nat(aTP_Lamm_aa(Uu),Uua),Uub) = heap_T3487192422709364219on_nat(aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,aa_nat_option_nat(some_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),Uua)),Uub)) ).

% ATP.lambda_620
tff(fact_10796_ATP_Olambda__621,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_vt(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub))) ).

% ATP.lambda_621
tff(fact_10797_ATP_Olambda__622,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vr(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub))) ).

% ATP.lambda_622
tff(fact_10798_ATP_Olambda__623,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_att(Uu),Uua),Uub) = aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),Uua)) ).

% ATP.lambda_623
tff(fact_10799_ATP_Olambda__624,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_aqc(Uu),Uua),Uub) = pure_assn(Uub = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uu),Uua)) ).

% ATP.lambda_624
tff(fact_10800_ATP_Olambda__625,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_aqd(Uu),Uua),Uub) = pure_assn(Uub = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uu),Uua)) ).

% ATP.lambda_625
tff(fact_10801_ATP_Olambda__626,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(aTP_Lamm_wb,Uu),Uua),Uub) = heap_T1489671670754571048_VEBTi(aTP_Lamm_sp) ).

% ATP.lambda_626
tff(fact_10802_ATP_Olambda__627,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat] : aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(aTP_Lamm_tx,Uu),Uua),Uub) = heap_Time_Heap_o2(aTP_Lamm_so) ).

% ATP.lambda_627
tff(fact_10803_ATP_Olambda__628,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(aTP_Lamm_rq,Uu),Uua),Uub) = heap_T5286843759275942675on_nat(aTP_Lamm_rp) ).

% ATP.lambda_628
tff(fact_10804_ATP_Olambda__629,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(aTP_Lamm_zs(Uu),Uua),Uub),Uuc) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_nat3009484633486537743_VEBTi(aa_VEB600817258366935214_VEBTi(aTP_Lamm_zr(Uu),Uua),Uuc),aTP_Lamm_xb(Uuc)),Uub) ).

% ATP.lambda_629
tff(fact_10805_ATP_Olambda__630,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(aTP_Lamm_xc(Uu),Uua),Uub),Uuc) = aa_VEB836174949890266638_VEBTi(vEBT_c6028912655521741485_VEBTi(aa_nat3009484633486537743_VEBTi(aa_VEB600817258366935214_VEBTi(aTP_Lamm_xa(Uu),Uua),Uuc),aTP_Lamm_xb(Uuc)),Uub) ).

% ATP.lambda_630
tff(fact_10806_ATP_Olambda__631,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(aTP_Lamm_tn(Uu),Uua),Uub),Uuc) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_nat1227502447850154924Heap_o(aa_VEB6636772557416003555Heap_o(aTP_Lamm_tm(Uu),Uua),Uuc),aTP_Lamm_nr(Uuc)),Uub) ).

% ATP.lambda_631
tff(fact_10807_ATP_Olambda__632,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(aTP_Lamm_py(Uu),Uua),Uub),Uuc) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_px(Uu),Uua),Uuc),aTP_Lamm_bk(Uuc)),Uub) ).

% ATP.lambda_632
tff(fact_10808_ATP_Olambda__633,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(aTP_Lamm_ph(Uu),Uua),Uub),Uuc) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_pg(Uu),Uua),Uuc),aTP_Lamm_cq(Uuc)),Uub) ).

% ATP.lambda_633
tff(fact_10809_ATP_Olambda__634,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(aTP_Lamm_ty(Uu),Uua),Uub),Uuc) = aa_VEB8194921341634505877Heap_o(vEBT_c6104975476656191286Heap_o(aa_nat1227502447850154924Heap_o(aa_VEB6636772557416003555Heap_o(aTP_Lamm_ol(Uu),Uua),Uuc),aTP_Lamm_nr(Uuc)),Uub) ).

% ATP.lambda_634
tff(fact_10810_ATP_Olambda__635,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(aTP_Lamm_rs(Uu),Uua),Uub),Uuc) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_gx(Uu),Uua),Uuc),aTP_Lamm_cq(Uuc)),Uub) ).

% ATP.lambda_635
tff(fact_10811_ATP_Olambda__636,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(aTP_Lamm_rr(Uu),Uua),Uub),Uuc) = aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_gi(Uu),Uua),Uuc),aTP_Lamm_bk(Uuc)),Uub) ).

% ATP.lambda_636
tff(fact_10812_ATP_Olambda__637,axiom,
    ! [Uu: array_VEBT_VEBTi,Uua: nat,Uub: nat,Uuc: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aTP_Lamm_nt(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uuc),heap_T3040810144269856602EBTi_o(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uu),Uua),aTP_Lamm_ns(Uub)),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false)) ).

% ATP.lambda_637
tff(fact_10813_ATP_Olambda__638,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      aa_VEBT_VEBT_real(aa_fun1825138017709814396T_real(aa_fun910392650362481850T_real(aTP_Lamm_ano(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_VEBT_VEBT_real(Uua,Uuc),aa_VEBT_VEBT_real(Uub,Uuc)) ).

% ATP.lambda_638
tff(fact_10814_ATP_Olambda__639,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: fun_VEBT_VEBT_rat,Uuc: vEBT_VEBT] :
      aa_VEBT_VEBT_rat(aa_fun8105832842879916156BT_rat(aa_fun2693410532875676774BT_rat(aTP_Lamm_anr(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_VEBT_VEBT_rat(Uua,Uuc),aa_VEBT_VEBT_rat(Uub,Uuc)) ).

% ATP.lambda_639
tff(fact_10815_ATP_Olambda__640,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_nat,Uub: fun_VEBT_VEBT_nat,Uuc: vEBT_VEBT] :
      aa_VEBT_VEBT_nat(aa_fun8284231930576791164BT_nat(aa_fun8778301808485728350BT_nat(aTP_Lamm_anu(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_VEBT_VEBT_nat(Uua,Uuc),aa_VEBT_VEBT_nat(Uub,Uuc)) ).

% ATP.lambda_640
tff(fact_10816_ATP_Olambda__641,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_int,Uub: fun_VEBT_VEBT_int,Uuc: vEBT_VEBT] :
      aa_VEBT_VEBT_int(aa_fun7144222995531728252BT_int(aa_fun7485955581636121530BT_int(aTP_Lamm_anx(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_VEBT_VEBT_int(Uua,Uuc),aa_VEBT_VEBT_int(Uub,Uuc)) ).

% ATP.lambda_641
tff(fact_10817_ATP_Olambda__642,axiom,
    ! [Uu: code_integer,Uua: fun_Co3960896333048504454r_real,Uub: fun_Co3960896333048504454r_real,Uuc: code_integer] :
      aa_Code_integer_real(aa_fun6341089502340155082r_real(aa_fun7644355603919855821r_real(aTP_Lamm_anq(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Code_integer_real(Uua,Uuc),aa_Code_integer_real(Uub,Uuc)) ).

% ATP.lambda_642
tff(fact_10818_ATP_Olambda__643,axiom,
    ! [Uu: code_integer,Uua: fun_Code_integer_rat,Uub: fun_Code_integer_rat,Uuc: code_integer] :
      aa_Code_integer_rat(aa_fun8169474305930283170er_rat(aa_fun1894662059565830609er_rat(aTP_Lamm_ant(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Code_integer_rat(Uua,Uuc),aa_Code_integer_rat(Uub,Uuc)) ).

% ATP.lambda_643
tff(fact_10819_ATP_Olambda__644,axiom,
    ! [Uu: code_integer,Uua: fun_Code_integer_nat,Uub: fun_Code_integer_nat,Uuc: code_integer] :
      aa_Code_integer_nat(aa_fun3076867170244285586er_nat(aa_fun7635586485349294009er_nat(aTP_Lamm_anw(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_Code_integer_nat(Uua,Uuc),aa_Code_integer_nat(Uub,Uuc)) ).

% ATP.lambda_644
tff(fact_10820_ATP_Olambda__645,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aa_fun8382785274855863872x_real(aTP_Lamm_anp(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_real(Uua,Uuc),aa_complex_real(Uub,Uuc)) ).

% ATP.lambda_645
tff(fact_10821_ATP_Olambda__646,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aa_fun530470884365435460ex_rat(aTP_Lamm_ans(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_rat(Uua,Uuc),aa_complex_rat(Uub,Uuc)) ).

% ATP.lambda_646
tff(fact_10822_ATP_Olambda__647,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: fun_complex_nat,Uuc: complex] :
      aa_complex_nat(aa_fun2465681967280144180ex_nat(aa_fun2484745150758625324ex_nat(aTP_Lamm_anv(Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa_complex_nat(Uua,Uuc),aa_complex_nat(Uub,Uuc)) ).

% ATP.lambda_647
tff(fact_10823_ATP_Olambda__648,axiom,
    ! [Uu: list_VEBT_VEBT,Uua: nat,Uub: nat,Uuc: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_nat1164146675577659234Heap_o(aa_nat3058146694327238959Heap_o(aTP_Lamm_nd(Uu),Uua),Uub),Uuc) = aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(vEBT_V854960066525838166emberi,nth_VEBT_VEBT(Uu,Uua)),Uuc),Uub) ).

% ATP.lambda_648
tff(fact_10824_ATP_Olambda__649,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: product_prod_nat_nat,Uuc: option_nat] :
      aa_opt9163293676379396302on_nat(aa_Pro4885747571561342816on_nat(aa_arr7927785801258559901on_nat(aTP_Lamm_ac(Uu),Uua),Uub),Uuc) = $ite(
        Uuc = none_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uub)),xa),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uub))),heap_T3487192422709364219on_nat(none_nat)),
        heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),the_nat(Uuc)),aa_opt1214731524852023858on_nat(aTP_Lamm_ab(Uu),Uuc)) ) ).

% ATP.lambda_649
tff(fact_10825_ATP_Olambda__650,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aTP_Lamm_ah(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uu),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uuc)),xa),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uuc))),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_Pro4465521767867336112on_nat(aa_VEB1870382382330055850on_nat(aa_arr666314619328815135on_nat(aTP_Lamm_ag(Uu),Uua),Uub),Uuc))) ) ).

% ATP.lambda_650
tff(fact_10826_ATP_Olambda__651,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: option_nat,Uuc: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_arr5509772438618118773on_nat(aTP_Lamm_ee(Uu),Uua),Uub),Uuc) = $ite(Uub = none_nat,heap_T3487192422709364219on_nat(none_nat),heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),the_nat(Uub)),aa_opt1214731524852023858on_nat(aTP_Lamm_ci(Uu),Uub))) ).

% ATP.lambda_651
tff(fact_10827_ATP_Olambda__652,axiom,
    ! [Uu: option4927543243414619207at_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aTP_Lamm_ai,Uu),Uua),Uub),Uuc) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aTP_Lamm_ah(Uua),Uub),Uuc),Uu) ).

% ATP.lambda_652
tff(fact_10828_ATP_Olambda__653,axiom,
    ! [Uu: $o,Uua: $o,Uub: code_integer,Uuc: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aTP_Lamm_asl((Uu)),(Uua)),Uub),(Uuc)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aTP_Lamm_ask((Uu)),(Uua)),(Uuc)),code_bit_cut_integer(Uub)) ).

% ATP.lambda_653
tff(fact_10829_ATP_Olambda__654,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: set_int,Uuc: int] : aa_int_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_aiq(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_aip(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_654
tff(fact_10830_ATP_Olambda__655,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: set_nat,Uuc: nat] : aa_nat_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_aio(Uu),Uua),Uub),Uuc) = groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_ain(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_655
tff(fact_10831_ATP_Olambda__656,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: set_complex,Uuc: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_aim(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_ail(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_656
tff(fact_10832_ATP_Olambda__657,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: set_nat,Uuc: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_aik(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_aij(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_657
tff(fact_10833_ATP_Olambda__658,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8905303033605938227omplex,Uub: fun_Co8975588572093281791plex_o,Uuc: complex] : aa_complex_complex(aa_fun7787660480616508700omplex(aa_fun963654016877434554omplex(aTP_Lamm_akv(Uu),Uua),Uub),Uuc) = groups8024822376189712711omplex(aa_com7552888802784664052omplex(aTP_Lamm_akt(Uua),Uuc),collect_Code_integer(aa_com4877492900066995176eger_o(aa_fun3025727905316630780eger_o(aTP_Lamm_aku(Uu),Uub),Uuc))) ).

% ATP.lambda_658
tff(fact_10834_ATP_Olambda__659,axiom,
    ! [Uu: set_int,Uua: fun_in1307110748760623106omplex,Uub: fun_in5065666251602127472plex_o,Uuc: complex] : aa_complex_complex(aa_fun7450476157521870733omplex(aa_fun6156949752960786844omplex(aTP_Lamm_akq(Uu),Uua),Uub),Uuc) = groups3049146728041665814omplex(aa_com9156536935073639875omplex(aTP_Lamm_ako(Uua),Uuc),collect_int(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_akp(Uu),Uub),Uuc))) ).

% ATP.lambda_659
tff(fact_10835_ATP_Olambda__660,axiom,
    ! [Uu: set_nat,Uua: fun_na1591161823130732966omplex,Uub: fun_na1554462459066888140plex_o,Uuc: complex] : aa_complex_complex(aa_fun1423577888295648489omplex(aa_fun601428455815558228omplex(aTP_Lamm_akl(Uu),Uua),Uub),Uuc) = groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_akj(Uua),Uuc),collect_nat(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_akk(Uu),Uub),Uuc))) ).

% ATP.lambda_660
tff(fact_10836_ATP_Olambda__661,axiom,
    ! [Uu: set_real,Uua: fun_re7208826350674946562omplex,Uub: fun_re7319081385389557872plex_o,Uuc: complex] : aa_complex_complex(aa_fun9084337531551611917omplex(aa_fun6739043078600750748omplex(aTP_Lamm_akg(Uu),Uua),Uub),Uuc) = groups5754745047067104278omplex(aa_com4766338864777664963omplex(aTP_Lamm_akf(Uua),Uuc),collect_real(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_adh(Uu),Uub),Uuc))) ).

% ATP.lambda_661
tff(fact_10837_ATP_Olambda__662,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE2006399705683203104omplex,Uub: fun_VE7631951351884037842plex_o,Uuc: complex] : aa_complex_complex(aa_fun2592672125884331625omplex(aa_fun8556514703313836300omplex(aTP_Lamm_akc(Uu),Uua),Uub),Uuc) = groups1794756597179926696omplex(aa_com2862003112617944841omplex(aTP_Lamm_akb(Uua),Uuc),collect_VEBT_VEBT(aa_com3651963455866140051VEBT_o(aa_fun7268111043052431330VEBT_o(aTP_Lamm_adg(Uu),Uub),Uuc))) ).

% ATP.lambda_662
tff(fact_10838_ATP_Olambda__663,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3312552843764780407at_nat,Uub: fun_Co1375048237525739357_nat_o,Uuc: nat] : aa_nat_nat(aa_fun7456160887942273982at_nat(aa_fun4417447297828429592at_nat(aTP_Lamm_ajy(Uu),Uua),Uub),Uuc) = groups7237345082560585321er_nat(aa_nat5852537043073606456er_nat(aTP_Lamm_ajw(Uua),Uuc),collect_Code_integer(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_ajx(Uu),Uub),Uuc))) ).

% ATP.lambda_663
tff(fact_10839_ATP_Olambda__664,axiom,
    ! [Uu: set_complex,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: nat] : aa_nat_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_ajt(Uu),Uua),Uub),Uuc) = groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_ajr(Uua),Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_ajs(Uu),Uub),Uuc))) ).

% ATP.lambda_664
tff(fact_10840_ATP_Olambda__665,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_ajo(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_ajm(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_ajn(Uu),Uub),Uuc))) ).

% ATP.lambda_665
tff(fact_10841_ATP_Olambda__666,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_ajj(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_aji(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_add(Uu),Uub),Uuc))) ).

% ATP.lambda_666
tff(fact_10842_ATP_Olambda__667,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE6480751511521678896_nat_o,Uuc: nat] : aa_nat_nat(aa_fun6249118799011721995at_nat(aa_fun2705500642357200114at_nat(aTP_Lamm_ajf(Uu),Uua),Uub),Uuc) = groups771621172384141258BT_nat(aa_nat7776146579971162249BT_nat(aTP_Lamm_aje(Uua),Uuc),collect_VEBT_VEBT(aa_nat1925708948643195573VEBT_o(aa_fun2405540735884763746VEBT_o(aTP_Lamm_adc(Uu),Uub),Uuc))) ).

% ATP.lambda_667
tff(fact_10843_ATP_Olambda__668,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_apc(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_apb(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).

% ATP.lambda_668
tff(fact_10844_ATP_Olambda__669,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ata(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_669
tff(fact_10845_ATP_Olambda__670,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_asy(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_670
tff(fact_10846_ATP_Olambda__671,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_asw(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uub),Uua)),Uuc)) ).

% ATP.lambda_671
tff(fact_10847_ATP_Olambda__672,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_asx(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uua),Uub)),Uuc)) ).

% ATP.lambda_672
tff(fact_10848_ATP_Olambda__673,axiom,
    ! [Uu: fun_num_option_num,Uua: fun_Pr6331313853612757382_num_o,Uub: num,Uuc: num] :
      ( aa_num_o(aa_num_fun_num_o(aa_fun1566934675525854290_num_o(aTP_Lamm_md(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_num_option_num(Uu,Uub) = aa_num_option_num(some_num,Uuc) )
        & aa_Pro7685007914727650815_num_o(Uua,aa_num3156681643174609981um_num(product_Pair_num_num(Uub),Uuc)) ) ) ).

% ATP.lambda_673
tff(fact_10849_ATP_Olambda__674,axiom,
    ! [Uu: fun_na5542100196373022448T_VEBT,Uua: fun_Pr6320724876807202603VEBT_o,Uub: nat,Uuc: produc4813437837504472865T_VEBT] :
      ( aa_Pro3461009536670225267VEBT_o(aa_nat6863640065291604488VEBT_o(aa_fun1244858531897282742VEBT_o(aTP_Lamm_mc(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_nat7129741047398451417T_VEBT(Uu,Uub) = some_P2407225848856114310T_VEBT(Uuc) )
        & aa_Pro1497393596385541012VEBT_o(Uua,aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(Uub),Uuc)) ) ) ).

% ATP.lambda_674
tff(fact_10850_ATP_Olambda__675,axiom,
    ! [Uu: fun_nat_option_num,Uua: fun_Pr5555124106300582480_num_o,Uub: nat,Uuc: num] :
      ( aa_num_o(aa_nat_fun_num_o(aa_fun8093532814315391442_num_o(aTP_Lamm_me(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_nat_option_num(Uu,Uub) = aa_num_option_num(some_num,Uuc) )
        & aa_Pro6908818167415475913_num_o(Uua,aa_num5026351239867140595at_num(product_Pair_nat_num(Uub),Uuc)) ) ) ).

% ATP.lambda_675
tff(fact_10851_ATP_Olambda__676,axiom,
    ! [Uu: fun_nat_option_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun2218514334412473170_nat_o(aTP_Lamm_mf(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_nat_option_nat(Uu,Uub) = aa_nat_option_nat(some_nat,Uuc) )
        & aa_Pro6150637604445964947_nat_o(Uua,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uub),Uuc)) ) ) ).

% ATP.lambda_676
tff(fact_10852_ATP_Olambda__677,axiom,
    ! [Uu: fun_int_option_int,Uua: fun_Pr3238374608701176162_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun5140975492127840594_int_o(aTP_Lamm_mg(Uu),Uua),Uub),Uuc)
    <=> ( ( aa_int_option_int(Uu,Uub) = aa_int_option_int(some_int,Uuc) )
        & aa_Pro4592068669816069595_int_o(Uua,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uub),Uuc)) ) ) ).

% ATP.lambda_677
tff(fact_10853_ATP_Olambda__678,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_aox(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_678
tff(fact_10854_ATP_Olambda__679,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: nat] : aa_nat2910048432523430448Heap_o(aa_arr494317875797454629Heap_o(aa_nat857754347168699366Heap_o(aTP_Lamm_nv(Uu),Uua),Uub),Uuc) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat5592791454563588861Heap_o(aTP_Lamm_nu(Uub),Uuc)) ).

% ATP.lambda_679
tff(fact_10855_ATP_Olambda__680,axiom,
    ! [Uu: set_complex,Uua: fun_VE7631951351884037842plex_o,Uub: vEBT_VEBT,Uuc: complex] :
      ( aa_complex_o(aa_VEB8036846806599048395plex_o(aa_fun5896829868240219618plex_o(aTP_Lamm_ajz(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_VEB8036846806599048395plex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_680
tff(fact_10856_ATP_Olambda__681,axiom,
    ! [Uu: set_complex,Uua: fun_Co8975588572093281791plex_o,Uub: code_integer,Uuc: complex] :
      ( aa_complex_o(aa_Cod7908732550325689320plex_o(aa_fun2498529521300939132plex_o(aTP_Lamm_akr(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_Cod7908732550325689320plex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_681
tff(fact_10857_ATP_Olambda__682,axiom,
    ! [Uu: set_complex,Uua: fun_re7319081385389557872plex_o,Uub: real,Uuc: complex] :
      ( aa_complex_o(aa_rea2382570592068293209plex_o(aa_fun789424914597169340plex_o(aTP_Lamm_akd(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_682
tff(fact_10858_ATP_Olambda__683,axiom,
    ! [Uu: set_complex,Uua: fun_na1554462459066888140plex_o,Uub: nat,Uuc: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_akh(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_nat_fun_complex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_683
tff(fact_10859_ATP_Olambda__684,axiom,
    ! [Uu: set_complex,Uua: fun_in5065666251602127472plex_o,Uub: int,Uuc: complex] :
      ( aa_complex_o(aa_int_fun_complex_o(aa_fun9089661398657494332plex_o(aTP_Lamm_akm(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_int_fun_complex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_684
tff(fact_10860_ATP_Olambda__685,axiom,
    ! [Uu: set_nat,Uua: fun_VE6480751511521678896_nat_o,Uub: vEBT_VEBT,Uuc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(aa_fun1998352179508271774_nat_o(aTP_Lamm_ajc(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_VEB2314620471664397097_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_685
tff(fact_10861_ATP_Olambda__686,axiom,
    ! [Uu: set_nat,Uua: fun_Co1375048237525739357_nat_o,Uub: code_integer,Uuc: nat] :
      ( aa_nat_o(aa_Cod1075288278386115654_nat_o(aa_fun2430183811687954940_nat_o(aTP_Lamm_aju(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_Cod1075288278386115654_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_686
tff(fact_10862_ATP_Olambda__687,axiom,
    ! [Uu: set_nat,Uua: fun_co5676208753217392844_nat_o,Uub: complex,Uuc: nat] :
      ( aa_nat_o(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_ajp(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_complex_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_687
tff(fact_10863_ATP_Olambda__688,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_o,Uub: real,Uuc: nat] :
      ( aa_nat_o(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_ajg(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_real_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_688
tff(fact_10864_ATP_Olambda__689,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_o,Uub: int,Uuc: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_ajk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_689
tff(fact_10865_ATP_Olambda__690,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE1909044406494083811eger_o,Uub: code_integer,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_Cod5040069908416755522VEBT_o(aa_fun1637734041313256866VEBT_o(aTP_Lamm_adi(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & aa_Code_integer_o(aa_VEB1409644639000594780eger_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_690
tff(fact_10866_ATP_Olambda__691,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE7631951351884037842plex_o,Uub: complex,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_com3651963455866140051VEBT_o(aa_fun7268111043052431330VEBT_o(aTP_Lamm_adg(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & aa_complex_o(aa_VEB8036846806599048395plex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_691
tff(fact_10867_ATP_Olambda__692,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE6480751511521678896_nat_o,Uub: nat,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(aa_fun2405540735884763746VEBT_o(aTP_Lamm_adc(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & aa_nat_o(aa_VEB2314620471664397097_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_692
tff(fact_10868_ATP_Olambda__693,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VE4235606543851232980_int_o,Uub: int,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_int8839263053405020433VEBT_o(aa_fun8807180201667926370VEBT_o(aTP_Lamm_ade(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & aa_int_o(aa_VEB493578126231709133_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_693
tff(fact_10869_ATP_Olambda__694,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co8975588572093281791plex_o,Uub: complex,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_com4877492900066995176eger_o(aa_fun3025727905316630780eger_o(aTP_Lamm_aku(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_complex_o(aa_Cod7908732550325689320plex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_694
tff(fact_10870_ATP_Olambda__695,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co1375048237525739357_nat_o,Uub: nat,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_nat7420681472272698950eger_o(aa_fun7373950593701497468eger_o(aTP_Lamm_ajx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & aa_nat_o(aa_Cod1075288278386115654_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_695
tff(fact_10871_ATP_Olambda__696,axiom,
    ! [Uu: set_complex,Uua: fun_co5676208753217392844_nat_o,Uub: nat,Uuc: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_ajs(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_nat_o(aa_complex_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_696
tff(fact_10872_ATP_Olambda__697,axiom,
    ! [Uu: set_real,Uua: fun_re3424108668441675905eger_o,Uub: code_integer,Uuc: real] :
      ( aa_real_o(aa_Cod1315720040876898922real_o(aa_fun4247632250071955964real_o(aTP_Lamm_adj(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & aa_Code_integer_o(aa_rea4090721512283939562eger_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_697
tff(fact_10873_ATP_Olambda__698,axiom,
    ! [Uu: set_real,Uua: fun_re7319081385389557872plex_o,Uub: complex,Uuc: real] :
      ( aa_real_o(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_adh(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_698
tff(fact_10874_ATP_Olambda__699,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_o,Uub: nat,Uuc: real] :
      ( aa_real_o(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_add(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & aa_nat_o(aa_real_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_699
tff(fact_10875_ATP_Olambda__700,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_o,Uub: int,Uuc: real] :
      ( aa_real_o(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_adf(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & aa_int_o(aa_real_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_700
tff(fact_10876_ATP_Olambda__701,axiom,
    ! [Uu: set_nat,Uua: fun_na1554462459066888140plex_o,Uub: complex,Uuc: nat] :
      ( aa_nat_o(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_akk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_complex_o(aa_nat_fun_complex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_701
tff(fact_10877_ATP_Olambda__702,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_adk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_702
tff(fact_10878_ATP_Olambda__703,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_o,Uub: int,Uuc: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_adl(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_int_o(aa_nat_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_703
tff(fact_10879_ATP_Olambda__704,axiom,
    ! [Uu: set_int,Uua: fun_in5065666251602127472plex_o,Uub: complex,Uuc: int] :
      ( aa_int_o(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_akp(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_complex_o(aa_int_fun_complex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_704
tff(fact_10880_ATP_Olambda__705,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_o,Uub: nat,Uuc: int] :
      ( aa_int_o(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_ajn(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_705
tff(fact_10881_ATP_Olambda__706,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_aow(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_706
tff(fact_10882_ATP_Olambda__707,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_aqx(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_707
tff(fact_10883_ATP_Olambda__708,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_aqu(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_708
tff(fact_10884_ATP_Olambda__709,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_uint32,Uub: fun_VEBT_VEBT_uint32,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun6272599145770227614VEBT_o(aa_fun1156574460560677430VEBT_o(aTP_Lamm_aeh(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & ( aa_uint32_uint32(times_times_uint32(aa_VEBT_VEBT_uint32(Uua,Uuc)),aa_VEBT_VEBT_uint32(Uub,Uuc)) != one_one_uint32 ) ) ) ).

% ATP.lambda_709
tff(fact_10885_ATP_Olambda__710,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun1965331558590768990VEBT_o(aa_fun5637434397837810486VEBT_o(aTP_Lamm_aet(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & ( aa_real_real(times_times_real(aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_710
tff(fact_10886_ATP_Olambda__711,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co1554742298434871494uint32,Uub: fun_Co1554742298434871494uint32,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_fun8926496376162642256eger_o(aa_fun1119956648780089443eger_o(aTP_Lamm_aer(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_uint32_uint32(times_times_uint32(aa_Cod8401701657498852871uint32(Uua,Uuc)),aa_Cod8401701657498852871uint32(Uub,Uuc)) != one_one_uint32 ) ) ) ).

% ATP.lambda_711
tff(fact_10887_ATP_Olambda__712,axiom,
    ! [Uu: set_complex,Uua: fun_complex_uint32,Uub: fun_complex_uint32,Uuc: complex] :
      ( aa_complex_o(aa_fun436548534304416686plex_o(aa_fun2751405167760845232plex_o(aTP_Lamm_aep(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_uint32_uint32(times_times_uint32(aa_complex_uint32(Uua,Uuc)),aa_complex_uint32(Uub,Uuc)) != one_one_uint32 ) ) ) ).

% ATP.lambda_712
tff(fact_10888_ATP_Olambda__713,axiom,
    ! [Uu: set_real,Uua: fun_real_uint32,Uub: fun_real_uint32,Uuc: real] :
      ( aa_real_o(aa_fun7556296611544133298real_o(aa_fun5572330763077102134real_o(aTP_Lamm_aej(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_uint32_uint32(times_times_uint32(aa_real_uint32(Uua,Uuc)),aa_real_uint32(Uub,Uuc)) != one_one_uint32 ) ) ) ).

% ATP.lambda_713
tff(fact_10889_ATP_Olambda__714,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_aev(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_real_real(times_times_real(aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_714
tff(fact_10890_ATP_Olambda__715,axiom,
    ! [Uu: set_nat,Uua: fun_nat_uint32,Uub: fun_nat_uint32,Uuc: nat] :
      ( aa_nat_o(aa_fun1182053767699555178_nat_o(aa_fun6957237400125433034_nat_o(aTP_Lamm_ael(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_uint32_uint32(times_times_uint32(aa_nat_uint32(Uua,Uuc)),aa_nat_uint32(Uub,Uuc)) != one_one_uint32 ) ) ) ).

% ATP.lambda_715
tff(fact_10891_ATP_Olambda__716,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_aex(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_real_real(times_times_real(aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_716
tff(fact_10892_ATP_Olambda__717,axiom,
    ! [Uu: set_int,Uua: fun_int_uint32,Uub: fun_int_uint32,Uuc: int] :
      ( aa_int_o(aa_fun6992502331602369202_int_o(aa_fun3369613587901038774_int_o(aTP_Lamm_aen(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_uint32_uint32(times_times_uint32(aa_int_uint32(Uua,Uuc)),aa_int_uint32(Uub,Uuc)) != one_one_uint32 ) ) ) ).

% ATP.lambda_717
tff(fact_10893_ATP_Olambda__718,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_aez(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_real_real(times_times_real(aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_718
tff(fact_10894_ATP_Olambda__719,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_real,Uub: fun_VEBT_VEBT_real,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun1965331558590768990VEBT_o(aa_fun5637434397837810486VEBT_o(aTP_Lamm_afb(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_VEBT_VEBT_real(Uua,Uuc)),aa_VEBT_VEBT_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_719
tff(fact_10895_ATP_Olambda__720,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_rat,Uub: fun_VEBT_VEBT_rat,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun5889122065779655690VEBT_o(aa_fun1978406156034293430VEBT_o(aTP_Lamm_afn(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uuc),Uu)
        & ( plus_plus_rat(aa_VEBT_VEBT_rat(Uua,Uuc),aa_VEBT_VEBT_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_720
tff(fact_10896_ATP_Olambda__721,axiom,
    ! [Uu: set_Code_integer,Uua: fun_Co3960896333048504454r_real,Uub: fun_Co3960896333048504454r_real,Uuc: code_integer] :
      ( aa_Code_integer_o(aa_fun3640226575523069072eger_o(aa_fun5110011637356352227eger_o(aTP_Lamm_afl(Uu),Uua),Uub),Uuc)
    <=> ( aa_set5643387332196575111eger_o(member_Code_integer(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_Code_integer_real(Uua,Uuc)),aa_Code_integer_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_721
tff(fact_10897_ATP_Olambda__722,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_afj(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_722
tff(fact_10898_ATP_Olambda__723,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_afd(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_723
tff(fact_10899_ATP_Olambda__724,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_afp(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_real_o(member_real(Uuc),Uu)
        & ( plus_plus_rat(aa_real_rat(Uua,Uuc),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_724
tff(fact_10900_ATP_Olambda__725,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_aff(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_725
tff(fact_10901_ATP_Olambda__726,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_afr(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( plus_plus_rat(aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_726
tff(fact_10902_ATP_Olambda__727,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_afh(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_727
tff(fact_10903_ATP_Olambda__728,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_aft(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( plus_plus_rat(aa_int_rat(Uua,Uuc),aa_int_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_728
tff(fact_10904_ATP_Olambda__729,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: $o] : aa_o_assn(aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_aqh(Uu),Uua),Uub),(Uuc)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),pure_assn((Uuc) = aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Uu),Uub))) ).

% ATP.lambda_729
tff(fact_10905_ATP_Olambda__730,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option_nat] : aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_aqi(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),pure_assn(Uuc = aa_nat_option_nat(vEBT_vebt_succ(Uu),Uub))) ).

% ATP.lambda_730
tff(fact_10906_ATP_Olambda__731,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option_nat] : aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_aqj(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),pure_assn(Uuc = aa_nat_option_nat(vEBT_vebt_pred(Uu),Uub))) ).

% ATP.lambda_731
tff(fact_10907_ATP_Olambda__732,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_vx(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uuc)) ).

% ATP.lambda_732
tff(fact_10908_ATP_Olambda__733,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_vy(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_733
tff(fact_10909_ATP_Olambda__734,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_apb(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_734
tff(fact_10910_ATP_Olambda__735,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex,Uuc: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_ail(Uu),Uua),Uub),Uuc) = aa_complex_complex(aa_com3342990543769683525omplex(times_times_complex,aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uuc)) ).

% ATP.lambda_735
tff(fact_10911_ATP_Olambda__736,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_ain(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uuc)) ).

% ATP.lambda_736
tff(fact_10912_ATP_Olambda__737,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_aij(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uuc)) ).

% ATP.lambda_737
tff(fact_10913_ATP_Olambda__738,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_aip(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uuc)) ).

% ATP.lambda_738
tff(fact_10914_ATP_Olambda__739,axiom,
    ! [Uu: set_complex,Uua: fun_Co8905303033605938227omplex,Uub: fun_Co8975588572093281791plex_o,Uuc: code_integer] : aa_Cod5621350732091595593omplex(aa_fun9008897831946212043omplex(aa_fun8319717443755388649omplex(aTP_Lamm_aks(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_Cod3397930797123873268omplex(Uua,Uuc),collect_complex(aa_Cod7908732550325689320plex_o(aa_fun2498529521300939132plex_o(aTP_Lamm_akr(Uu),Uub),Uuc))) ).

% ATP.lambda_739
tff(fact_10915_ATP_Olambda__740,axiom,
    ! [Uu: set_complex,Uua: fun_in1307110748760623106omplex,Uub: fun_in5065666251602127472plex_o,Uuc: int] : aa_int_complex(aa_fun3141780957025892107omplex(aa_fun323909820831547930omplex(aTP_Lamm_akn(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_int3545933452204062147omplex(Uua,Uuc),collect_complex(aa_int_fun_complex_o(aa_fun9089661398657494332plex_o(aTP_Lamm_akm(Uu),Uub),Uuc))) ).

% ATP.lambda_740
tff(fact_10916_ATP_Olambda__741,axiom,
    ! [Uu: set_complex,Uua: fun_na1591161823130732966omplex,Uub: fun_na1554462459066888140plex_o,Uuc: nat] : aa_nat_complex(aa_fun4713049822795992587omplex(aa_fun3479084698764006518omplex(aTP_Lamm_aki(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(Uua,Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_akh(Uu),Uub),Uuc))) ).

% ATP.lambda_741
tff(fact_10917_ATP_Olambda__742,axiom,
    ! [Uu: set_complex,Uua: fun_re7208826350674946562omplex,Uub: fun_re7319081385389557872plex_o,Uuc: real] : aa_real_complex(aa_fun401690246322682763omplex(aa_fun7501666206714588442omplex(aTP_Lamm_ake(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_rea2618301233319580611omplex(Uua,Uuc),collect_complex(aa_rea2382570592068293209plex_o(aa_fun789424914597169340plex_o(aTP_Lamm_akd(Uu),Uub),Uuc))) ).

% ATP.lambda_742
tff(fact_10918_ATP_Olambda__743,axiom,
    ! [Uu: set_complex,Uua: fun_VE2006399705683203104omplex,Uub: fun_VE7631951351884037842plex_o,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_complex(aa_fun1953711888481404901omplex(aa_fun5829985485761010584omplex(aTP_Lamm_aka(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_VEB5608662635949521omplex(Uua,Uuc),collect_complex(aa_VEB8036846806599048395plex_o(aa_fun5896829868240219618plex_o(aTP_Lamm_ajz(Uu),Uub),Uuc))) ).

% ATP.lambda_743
tff(fact_10919_ATP_Olambda__744,axiom,
    ! [Uu: set_nat,Uua: fun_Co3312552843764780407at_nat,Uub: fun_Co1375048237525739357_nat_o,Uuc: code_integer] : aa_Code_integer_nat(aa_fun6683192525214198859er_nat(aa_fun2143651634668453157er_nat(aTP_Lamm_ajv(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_Cod8670293052248272184at_nat(Uua,Uuc)),collect_nat(aa_Cod1075288278386115654_nat_o(aa_fun2430183811687954940_nat_o(aTP_Lamm_aju(Uu),Uub),Uuc))) ).

% ATP.lambda_744
tff(fact_10920_ATP_Olambda__745,axiom,
    ! [Uu: set_nat,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: complex] : aa_complex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_ajq(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_ajp(Uu),Uub),Uuc))) ).

% ATP.lambda_745
tff(fact_10921_ATP_Olambda__746,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: int] : aa_int_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_ajl(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_ajk(Uu),Uub),Uuc))) ).

% ATP.lambda_746
tff(fact_10922_ATP_Olambda__747,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_ajh(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_ajg(Uu),Uub),Uuc))) ).

% ATP.lambda_747
tff(fact_10923_ATP_Olambda__748,axiom,
    ! [Uu: set_nat,Uua: fun_VE597728521808591204at_nat,Uub: fun_VE6480751511521678896_nat_o,Uuc: vEBT_VEBT] : aa_VEBT_VEBT_nat(aa_fun3621025987637128233BT_nat(aa_fun1603736613815373464BT_nat(aTP_Lamm_ajd(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_VEB1122092716774696213at_nat(Uua,Uuc)),collect_nat(aa_VEB2314620471664397097_nat_o(aa_fun1998352179508271774_nat_o(aTP_Lamm_ajc(Uu),Uub),Uuc))) ).

% ATP.lambda_748
tff(fact_10924_ATP_Olambda__749,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: option_nat] : aa_option_nat_assn(aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_aql(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,pure_assn(Uuc = vEBT_vebt_mint(nth_VEBT_VEBT(Uua,Uu)))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uua,Uub)) ).

% ATP.lambda_749
tff(fact_10925_ATP_Olambda__750,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: option_nat] : aa_option_nat_assn(aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_aqk(Uu),Uua),Uub),Uuc) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,pure_assn(Uuc = vEBT_vebt_maxt(nth_VEBT_VEBT(Uua,Uu)))),vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uua,Uub)) ).

% ATP.lambda_750
tff(fact_10926_ATP_Olambda__751,axiom,
    ! [Uu: option4927543243414619207at_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aTP_Lamm_bc,Uu),Uua),Uub),Uuc) = heap_T5999496708990702694on_nat(refine_Imp_assert(aa_VEBT_VEBT_o(vEBT_is_Node,ta)),aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aTP_Lamm_bb(Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_751
tff(fact_10927_ATP_Olambda__752,axiom,
    ! [Uu: fun_Pr238758252084143515Heap_o,Uua: array_VEBT_VEBTi,Uub: nat,Uuc: nat] : aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aTP_Lamm_st(Uu),Uua),Uub),Uuc) = heap_Time_bind_nat_o(array_len_VEBT_VEBTi(Uua),aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_arr5803378942169532735Heap_o(aTP_Lamm_ss(Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_752
tff(fact_10928_ATP_Olambda__753,axiom,
    ! [Uu: fun_VE7134199479003659425Heap_o,Uua: array_VEBT_VEBTi,Uub: nat,Uuc: nat] : aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aTP_Lamm_oo(Uu),Uua),Uub),Uuc) = heap_Time_bind_nat_o(array_len_VEBT_VEBTi(Uua),aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_arr5803378942169532735Heap_o(aTP_Lamm_on(Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_753
tff(fact_10929_ATP_Olambda__754,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_VEBT_VEBTi,Uuc: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_lis4026867407995182650_VEBTi(aa_nat1561540599483812487_VEBTi(aTP_Lamm_aaw(Uu),Uua),Uub),Uuc) = heap_T5099337393651448672_VEBTi(array_615059503499738533_VEBTi(Uub),aa_nat106358831840921632_VEBTi(aTP_Lamm_aav(Uu),Uua)) ).

% ATP.lambda_754
tff(fact_10930_ATP_Olambda__755,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_VEBT_VEBTi,Uuc: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_lis4026867407995182650_VEBTi(aa_nat1561540599483812487_VEBTi(aTP_Lamm_aat(Uu),Uua),Uub),Uuc) = heap_T5099337393651448672_VEBTi(array_615059503499738533_VEBTi(Uub),aa_nat106358831840921632_VEBTi(aTP_Lamm_aas(Uu),Uua)) ).

% ATP.lambda_755
tff(fact_10931_ATP_Olambda__756,axiom,
    ! [Uu: $o,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_hk((Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_756
tff(fact_10932_ATP_Olambda__757,axiom,
    ! [Uu: $o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_hl((Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_757
tff(fact_10933_ATP_Olambda__758,axiom,
    ! [Uu: option4927543243414619207at_nat,Uua: nat,Uub: list_VEBT_VEBT,Uuc: vEBT_VEBT] : aa_VEB8692087746923902260T_VEBT(aa_lis2966823273985285125T_VEBT(aa_nat6705503460730637082T_VEBT(aa_opt2902932101092860132T_VEBT(aTP_Lamm_az,Uu),Uua),Uub),Uuc) = produc6272728841373537334T_VEBT(Uu,aa_Pro7921459969303816107T_VEBT(produc1750349459881913976T_VEBT(Uua),produc6691630295680060561T_VEBT(Uub,Uuc))) ).

% ATP.lambda_758
tff(fact_10934_ATP_Olambda__759,axiom,
    ! [Uu: fun_He8247729123535230979_VEBTi,Uua: fun_o_6972675622308511895_VEBTi,Uub: $o,Uuc: $o] : aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(aa_fun5654756598358634620_VEBTi(aTP_Lamm_kl(Uu),Uua),(Uub)),(Uuc)) = aa_Hea2545666323642607868_VEBTi(Uu,aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_759
tff(fact_10935_ATP_Olambda__760,axiom,
    ! [Uu: fun_He1783186893022644846on_nat,Uua: fun_o_6972675622308511895_VEBTi,Uub: $o,Uuc: $o] : aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(aa_fun4045834922583578215on_nat(aTP_Lamm_kh(Uu),Uua),(Uub)),(Uuc)) = aa_Hea6259128614175166183on_nat(Uu,aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_760
tff(fact_10936_ATP_Olambda__761,axiom,
    ! [Uu: fun_He4624006784692404214Heap_o,Uua: fun_o_6972675622308511895_VEBTi,Uub: $o,Uuc: $o] : aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(aa_fun3457972572965721299Heap_o(aTP_Lamm_kj(Uu),Uua),(Uub)),(Uuc)) = aa_Hea7612170549154561063Heap_o(Uu,aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_761
tff(fact_10937_ATP_Olambda__762,axiom,
    ! [Uu: fun_He2623223239018661208_VEBTi,Uua: fun_o_4049874799390722306on_nat,Uub: $o,Uuc: $o] : aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(aa_fun8772555301277947729_VEBTi(aTP_Lamm_jz(Uu),Uua),(Uub)),(Uuc)) = aa_Hea2832269215923769937_VEBTi(Uu,aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_762
tff(fact_10938_ATP_Olambda__763,axiom,
    ! [Uu: fun_He8640631688564546883on_nat,Uua: fun_o_4049874799390722306on_nat,Uub: $o,Uuc: $o] : aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(aa_fun4373765338990682812on_nat(aTP_Lamm_jv(Uu),Uua),(Uub)),(Uuc)) = aa_Hea6597239020403563964on_nat(Uu,aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_763
tff(fact_10939_ATP_Olambda__764,axiom,
    ! [Uu: fun_He5988046180898962017Heap_o,Uua: fun_o_4049874799390722306on_nat,Uub: $o,Uuc: $o] : aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(aa_fun4676301296891262398Heap_o(aTP_Lamm_jx(Uu),Uua),(Uub)),(Uuc)) = aa_Hea662945835469240978Heap_o(Uu,aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_764
tff(fact_10940_ATP_Olambda__765,axiom,
    ! [Uu: fun_He6517945810935171576_VEBTi,Uua: fun_o_1207901889202581752Heap_o,Uub: $o,Uuc: $o] : aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(aa_fun8320297987776248033_VEBTi(aTP_Lamm_kf(Uu),Uua),(Uub)),(Uuc)) = aa_Hea4465279032261640929_VEBTi(Uu,aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_765
tff(fact_10941_ATP_Olambda__766,axiom,
    ! [Uu: fun_He6140748070426431971on_nat,Uua: fun_o_1207901889202581752Heap_o,Uub: $o,Uuc: $o] : aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(aa_fun3614415378023021644on_nat(aTP_Lamm_kb(Uu),Uua),(Uub)),(Uuc)) = aa_Hea41712926108436044on_nat(Uu,aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_766
tff(fact_10942_ATP_Olambda__767,axiom,
    ! [Uu: fun_He6597989805181038017Heap_o,Uua: fun_o_1207901889202581752Heap_o,Uub: $o,Uuc: $o] : aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(aa_fun3804294760359788590Heap_o(aTP_Lamm_kd(Uu),Uua),(Uub)),(Uuc)) = aa_Hea4824394052061459970Heap_o(Uu,aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_767
tff(fact_10943_ATP_Olambda__768,axiom,
    ! [Uu: fun_Pr3650993782763339253T_VEBT,Uua: fun_o_8142510167905321106T_VEBT,Uub: $o,Uuc: $o] : aa_o_P8223745691486049304T_VEBT(aa_o_f7062522455407347339T_VEBT(aa_fun5380946117177639010T_VEBT(aTP_Lamm_ij(Uu),Uua),(Uub)),(Uuc)) = aa_Pro5421721056952420150T_VEBT(Uu,aa_o_P8223745691486049304T_VEBT(aa_o_f7062522455407347339T_VEBT(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_768
tff(fact_10944_ATP_Olambda__769,axiom,
    ! [Uu: fun_Pr7471953115884902709VEBT_o,Uua: fun_o_8142510167905321106T_VEBT,Uub: $o,Uuc: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_fun449888776569364894un_o_o(aTP_Lamm_il(Uu),Uua),(Uub)),(Uuc))
    <=> aa_Pro5475924895142249886VEBT_o(Uu,aa_o_P8223745691486049304T_VEBT(aa_o_f7062522455407347339T_VEBT(Uua,(Uub)),(Uuc))) ) ).

% ATP.lambda_769
tff(fact_10945_ATP_Olambda__770,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: fun_na4138050610385705875at_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun7288258519458182931_nat_o(aTP_Lamm_cw(Uu),Uua),Uub),Uuc)
    <=> aa_Pro6150637604445964947_nat_o(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(Uua,Uub),Uuc)) ) ).

% ATP.lambda_770
tff(fact_10946_ATP_Olambda__771,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_cy(Uu),Uua),Uub),Uuc)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_771
tff(fact_10947_ATP_Olambda__772,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun7461304428735181397nt_int(aTP_Lamm_cz(Uu),Uua),Uub),Uuc) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_772
tff(fact_10948_ATP_Olambda__773,axiom,
    ! [Uu: fun_o_3001263932818058983T_VEBT,Uua: fun_o_fun_o_o,Uub: $o,Uuc: $o] : aa_o_P8223745691486049304T_VEBT(aa_o_f7062522455407347339T_VEBT(aa_fun4185041498930275652T_VEBT(aTP_Lamm_in(Uu),Uua),(Uub)),(Uuc)) = aa_o_P8223745691486049304T_VEBT(Uu,aa_o_o(aa_o_fun_o_o(Uua,(Uub)),(Uuc))) ).

% ATP.lambda_773
tff(fact_10949_ATP_Olambda__774,axiom,
    ! [Uu: fun_o_67843798866261420at_nat,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_fun9180491248050787109at_nat(aTP_Lamm_cx(Uu),Uua),Uub),Uuc) = aa_o_P6312935399938478757at_nat(Uu,aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_774
tff(fact_10950_ATP_Olambda__775,axiom,
    ! [Uu: fun_o_8988054265482332260nt_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_da(Uu),Uua),Uub),Uuc) = aa_o_P3134438092330336605nt_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_775
tff(fact_10951_ATP_Olambda__776,axiom,
    ! [Uu: fun_o_o,Uua: fun_o_fun_o_o,Uub: $o,Uuc: $o] :
      ( aa_o_o(aa_o_fun_o_o(aa_fun6846671739589424444un_o_o(aTP_Lamm_ip(Uu),Uua),(Uub)),(Uuc))
    <=> aa_o_o(Uu,aa_o_o(aa_o_fun_o_o(Uua,(Uub)),(Uuc))) ) ).

% ATP.lambda_776
tff(fact_10952_ATP_Olambda__777,axiom,
    ! [Uu: fun_o_o,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_cr(Uu),Uua),Uub),Uuc)
    <=> aa_o_o(Uu,aa_nat_o(aa_nat_fun_nat_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_777
tff(fact_10953_ATP_Olambda__778,axiom,
    ! [Uu: fun_o_o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_cs(Uu),Uua),Uub),Uuc)
    <=> aa_o_o(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_778
tff(fact_10954_ATP_Olambda__779,axiom,
    ! [Uu: fun_o_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun7323606162024495668nt_int(aTP_Lamm_ct(Uu),Uua),Uub),Uuc) = aa_o_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_779
tff(fact_10955_ATP_Olambda__780,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun82482788810740534_int_o(aTP_Lamm_cu(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_780
tff(fact_10956_ATP_Olambda__781,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun6119123525526402106nt_int(aTP_Lamm_cv(Uu),Uua),Uub),Uuc) = aa_int_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_781
tff(fact_10957_ATP_Olambda__782,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_ani(Uu),Uua),Uub),Uuc) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_782
tff(fact_10958_ATP_Olambda__783,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_anh(Uu),Uua),Uub),Uuc) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_783
tff(fact_10959_ATP_Olambda__784,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_arb(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_784
tff(fact_10960_ATP_Olambda__785,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_aqz(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_785
tff(fact_10961_ATP_Olambda__786,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_arf(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uub)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_786
tff(fact_10962_ATP_Olambda__787,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ard(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).

% ATP.lambda_787
tff(fact_10963_ATP_Olambda__788,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: list_VEBT_VEBT,Uub: nat,Uuc: nat,Uud: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_nat1164146675577659234Heap_o(aa_nat3058146694327238959Heap_o(aa_lis6732064841504156908Heap_o(aTP_Lamm_sz(Uu),Uua),Uub),Uuc),Uud) = aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(produc8906325494159015119Heap_o(produc8381543706267210711Heap_o(Uu)),nth_VEBT_VEBT(Uua,Uub)),Uud),Uuc) ).

% ATP.lambda_788
tff(fact_10964_ATP_Olambda__789,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: list_VEBT_VEBT,Uub: nat,Uuc: nat,Uud: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_nat1164146675577659234Heap_o(aa_nat3058146694327238959Heap_o(aa_lis6732064841504156908Heap_o(aTP_Lamm_ny(Uu),Uua),Uub),Uuc),Uud) = aa_nat2910048432523430448Heap_o(aa_VEB1455453220190609250Heap_o(aa_VEB8672304581938048793Heap_o(Uu,nth_VEBT_VEBT(Uua,Uub)),Uud),Uuc) ).

% ATP.lambda_789
tff(fact_10965_ATP_Olambda__790,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi] :
      aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aTP_Lamm_yr(Uu),Uua),Uub),Uuc),Uud) = case_o1356590567247012107at_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uua,Uub,Uuc),Uud)),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uu),Uu)),Uub,Uuc),Uud))),aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aTP_Lamm_yq(Uu),Uua),Uub),Uuc),Uud),Uua) ).

% ATP.lambda_790
tff(fact_10966_ATP_Olambda__791,axiom,
    ! [Uu: fun_Pr238758252084143515Heap_o,Uua: array_VEBT_VEBTi,Uub: nat,Uuc: nat,Uud: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_arr5803378942169532735Heap_o(aTP_Lamm_ss(Uu),Uua),Uub),Uuc),Uud) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uud),heap_T3040810144269856602EBTi_o(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),Uub),aa_nat1164146675577659234Heap_o(aTP_Lamm_sr(Uu),Uuc)),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false)) ).

% ATP.lambda_791
tff(fact_10967_ATP_Olambda__792,axiom,
    ! [Uu: fun_VE7134199479003659425Heap_o,Uua: array_VEBT_VEBTi,Uub: nat,Uuc: nat,Uud: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_arr5803378942169532735Heap_o(aTP_Lamm_on(Uu),Uua),Uub),Uuc),Uud) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uud),heap_T3040810144269856602EBTi_o(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),Uub),aa_nat1164146675577659234Heap_o(aTP_Lamm_om(Uu),Uuc)),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false)) ).

% ATP.lambda_792
tff(fact_10968_ATP_Olambda__793,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: product_prod_nat_nat,Uud: option_nat] :
      aa_opt9163293676379396302on_nat(aa_Pro4885747571561342816on_nat(aa_arr7927785801258559901on_nat(aa_nat6148929329942773342on_nat(aTP_Lamm_bd(Uu),Uua),Uub),Uuc),Uud) = $ite(
        Uud = none_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uuc)),Uu),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uuc))),heap_T3487192422709364219on_nat(none_nat)),
        heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),the_nat(Uud)),aa_opt1214731524852023858on_nat(aTP_Lamm_ab(Uua),Uud)) ) ).

% ATP.lambda_793
tff(fact_10969_ATP_Olambda__794,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: product_prod_nat_nat,Uuc: option_nat,Uud: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_Pro5009687527940591364on_nat(aa_arr722815569820881053on_nat(aTP_Lamm_ak(Uu),Uua),Uub),Uuc),Uud) = $ite(
        Uuc = none_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uub)),xa),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uub))),heap_T3487192422709364219on_nat(none_nat)),
        heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),the_nat(Uuc)),aa_opt1214731524852023858on_nat(aTP_Lamm_ab(Uu),Uuc)) ) ).

% ATP.lambda_794
tff(fact_10970_ATP_Olambda__795,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aTP_Lamm_bi(Uu),Uua),Uub),Uuc),Uud) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uud)),Uu),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uud))),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_Pro4465521767867336112on_nat(aa_VEB1870382382330055850on_nat(aa_arr666314619328815135on_nat(aa_nat8253036615571765984on_nat(aTP_Lamm_bh(Uu),Uua),Uub),Uuc),Uud))) ) ).

% ATP.lambda_795
tff(fact_10971_ATP_Olambda__796,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aTP_Lamm_co(Uu),Uua),Uub),Uuc),Uud) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uud)),
          heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uud))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uud)),Uu),heap_T3487192422709364219on_nat(none_nat),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_VEB3566417204837707778on_nat(aa_arr5230852142804057699on_nat(aa_nat6371187903253094776on_nat(aTP_Lamm_cn(Uu),Uua),Uub),Uuc))) ) ) ).

% ATP.lambda_796
tff(fact_10972_ATP_Olambda__797,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: nat,Uud: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aTP_Lamm_nw(Uu),Uua),Uub),Uuc),Uud) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),one_one_nat),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
        $ite(
          Uu = Uuc,
          aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
          $ite(
            Uu = Uud,
            aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),Uuc),
              aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uud),Uu),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_arr494317875797454629Heap_o(aa_nat857754347168699366Heap_o(aTP_Lamm_nv(Uu),Uua),Uub))) ) ) ) ) ).

% ATP.lambda_797
tff(fact_10973_ATP_Olambda__798,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aTP_Lamm_cp(Uu),Uua),Uub),Uuc),Uud) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aTP_Lamm_co(Uu),Uub),Uuc),Uud),Uua) ).

% ATP.lambda_798
tff(fact_10974_ATP_Olambda__799,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aTP_Lamm_bj(Uu),Uua),Uub),Uuc),Uud) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aTP_Lamm_bi(Uu),Uub),Uuc),Uud),Uua) ).

% ATP.lambda_799
tff(fact_10975_ATP_Olambda__800,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aTP_Lamm_nx(Uu),Uua),Uub),Uuc),Uud) = case_o1442776274061689234at_nat(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc3505292621261808240Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aTP_Lamm_nw(Uu),Uub),Uuc)),Uua) ).

% ATP.lambda_800
tff(fact_10976_ATP_Olambda__801,axiom,
    ! [Uu: option4927543243414619207at_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aTP_Lamm_bb(Uu),Uua),Uub),Uuc),Uud) = produc2190226783763740553on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aTP_Lamm_ay(Uu),Uua),Uub),Uuc),vEBT_c634343235235684882T_VEBT(aTP_Lamm_az,aTP_Lamm_ba,ta)) ).

% ATP.lambda_801
tff(fact_10977_ATP_Olambda__802,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: code_integer,Uud: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aTP_Lamm_ask((Uu)),(Uua)),(Uub)),Uuc),(Uud)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aTP_Lamm_asj((Uu)),(Uua)),(Uub)),(Uud)),code_bit_cut_integer(Uuc)) ).

% ATP.lambda_802
tff(fact_10978_ATP_Olambda__803,axiom,
    ! [Uu: nat,Uua: fun_na6724076748692969250l_real,Uub: real,Uuc: nat,Uud: real] : aa_real_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_asz(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_asy(Uua),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(times_times_real(Uub),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_803
tff(fact_10979_ATP_Olambda__804,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: nat] : aa_nat2910048432523430448Heap_o(aa_lis5083133406047025005Heap_o(aa_arr6185401592279617542Heap_o(aa_nat1466035401240590747Heap_o(aTP_Lamm_nj(Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat5592791454563588861Heap_o(aa_lis455817436297872826Heap_o(aa_arr1217223296832173011Heap_o(aa_nat6014765364524370664Heap_o(aTP_Lamm_ni(Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_804
tff(fact_10980_ATP_Olambda__805,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: product_unit] : aa_Pro3379321066291785715Heap_o(aa_lis4443831354813659076Heap_o(aa_arr8974760139355794305Heap_o(aa_nat808430574047188546Heap_o(aTP_Lamm_nk(Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_lis5083133406047025005Heap_o(aa_arr6185401592279617542Heap_o(aa_nat1466035401240590747Heap_o(aTP_Lamm_nj(Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_805
tff(fact_10981_ATP_Olambda__806,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_arr5230852142804057699on_nat(aa_nat6371187903253094776on_nat(aTP_Lamm_cn(Uu),Uua),Uub),Uuc),Uud) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_VEB6424020099669959819on_nat(aa_arr7085280839797930604on_nat(aTP_Lamm_cm(Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_806
tff(fact_10982_ATP_Olambda__807,axiom,
    ! [Uu: fun_Pr238758252084143515Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: nat] : aa_nat2910048432523430448Heap_o(aa_arr494317875797454629Heap_o(aa_nat857754347168699366Heap_o(aa_nat8612037029889215215Heap_o(aTP_Lamm_su(Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aTP_Lamm_st(Uu),Uuc),Uud)) ).

% ATP.lambda_807
tff(fact_10983_ATP_Olambda__808,axiom,
    ! [Uu: fun_VE7134199479003659425Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: nat] : aa_nat2910048432523430448Heap_o(aa_arr494317875797454629Heap_o(aa_nat857754347168699366Heap_o(aa_nat8612037029889215215Heap_o(aTP_Lamm_op(Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aTP_Lamm_oo(Uu),Uuc),Uud)) ).

% ATP.lambda_808
tff(fact_10984_ATP_Olambda__809,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_VEB6424020099669959819on_nat(aa_arr7085280839797930604on_nat(aTP_Lamm_cm(Uu),Uua),Uub),Uuc),Uud) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),Uud),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB216942459957533935on_nat(aa_arr2558136924908760784on_nat(aTP_Lamm_cl(Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_809
tff(fact_10985_ATP_Olambda__810,axiom,
    ! [Uu: array_VEBT_VEBTi,Uua: list_VEBT_VEBT,Uub: nat,Uuc: nat,Uud: product_unit] : aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aTP_Lamm_ne(Uu),Uua),Uub),Uuc),Uud) = heap_T3040810144269856602EBTi_o(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uu),Uub),aa_nat1164146675577659234Heap_o(aa_nat3058146694327238959Heap_o(aTP_Lamm_nd(Uua),Uub),Uuc)) ).

% ATP.lambda_810
tff(fact_10986_ATP_Olambda__811,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: product_prod_nat_nat,Uud: nat] : aa_nat947733127081326238on_nat(aa_Pro4465521767867336112on_nat(aa_VEB1870382382330055850on_nat(aa_arr666314619328815135on_nat(aTP_Lamm_ag(Uu),Uua),Uub),Uuc),Uud) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB358289369652911795on_nat(aa_arr4041137963511513512on_nat(aTP_Lamm_af(Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_811
tff(fact_10987_ATP_Olambda__812,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBT,Uuc: nat,Uud: option_nat] :
      aa_opt9163293676379396302on_nat(aa_nat9127621225095157975on_nat(aa_VEB8264456441901112694on_nat(aa_arr7060518384107407923on_nat(aTP_Lamm_ef(Uu),Uua),Uub),Uuc),Uud) = heap_T5999496708990702694on_nat(refine_Imp_assert(( ( Uud = none_nat )
          = ( aa_nat_option_nat(vEBT_vebt_succ(Uub),Uuc) = none_nat ) )),aa_opt5089197864053947520on_nat(aa_arr5509772438618118773on_nat(aTP_Lamm_ee(Uu),Uua),Uud)) ).

% ATP.lambda_812
tff(fact_10988_ATP_Olambda__813,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: option4927543243414619207at_nat] : aa_opt6266093870483638640Heap_o(aa_arr6198139058929667337Heap_o(aa_nat436397609992601886Heap_o(aa_opt4438414422311449448Heap_o(aTP_Lamm_nn(Uu),Uua),Uub),Uuc),Uud) = produc5946672270950774087Heap_o(aa_opt7844807959364121492Heap_o(aa_arr79663438428320465Heap_o(aa_nat2087714105643485586Heap_o(aa_opt9028207981337828190Heap_o(aTP_Lamm_nm(Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_813
tff(fact_10989_ATP_Olambda__814,axiom,
    ! [Uu: option4927543243414619207at_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: option4927543243414619207at_nat] : aa_opt1657869255803490932on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aTP_Lamm_ay(Uu),Uua),Uub),Uuc),Uud) = produc2606485630176857543on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aTP_Lamm_ax(Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_814
tff(fact_10990_ATP_Olambda__815,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat] : aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_arr6831033733615195768_VEBTi(aa_nat7652049589537018893_VEBTi(aTP_Lamm_wc(Uu),Uua),Uub),Uuc),Uud) = aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uud)),Uu,Uub),Uuc)) ).

% ATP.lambda_815
tff(fact_10991_ATP_Olambda__816,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] :
      aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aa_nat3009484633486537743_VEBTi(aTP_Lamm_aaq(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o1356590567247012107at_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uuc),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uub,Uuc,Uud),Uue)),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uua)),Uuc,Uud),Uue))),aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aTP_Lamm_aap(Uu),Uua),Uub),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_816
tff(fact_10992_ATP_Olambda__817,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] :
      aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aa_nat3009484633486537743_VEBTi(aTP_Lamm_ye(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o1356590567247012107at_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uuc),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uub,Uuc,Uud),Uue)),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uua)),Uuc,Uud),Uue))),aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aTP_Lamm_yd(Uu),Uua),Uub),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_817
tff(fact_10993_ATP_Olambda__818,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] :
      aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aa_nat3009484633486537743_VEBTi(aTP_Lamm_xp(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o1356590567247012107at_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uuc),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uub,Uuc,Uud),Uue)),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uua)),Uuc,Uud),Uue))),aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aTP_Lamm_xo(Uu),Uua),Uub),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_818
tff(fact_10994_ATP_Olambda__819,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: product_prod_nat_nat,Uud: option_nat,Uue: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_Pro5009687527940591364on_nat(aa_arr722815569820881053on_nat(aa_nat5945554870259702194on_nat(aTP_Lamm_bl(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        Uud = none_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uuc)),Uu),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uuc))),heap_T3487192422709364219on_nat(none_nat)),
        heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),the_nat(Uud)),aa_opt1214731524852023858on_nat(aTP_Lamm_ab(Uua),Uud)) ) ).

% ATP.lambda_819
tff(fact_10995_ATP_Olambda__820,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat] :
      aa_Pro1395359565518501564_VEBTi(aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aTP_Lamm_yq(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uua,Uub,Uuc),Uud)),heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uue)),aa_Pro3150526172571800645_VEBTi(aa_VEB4890588985164552255_VEBTi(aa_arr5592147402706875956_VEBTi(aa_nat5963232373128642933_VEBTi(aTP_Lamm_yp(Uu),Uub),Uuc),Uud),Uue))) ).

% ATP.lambda_820
tff(fact_10996_ATP_Olambda__821,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aa_nat8243565228001313504on_nat(aTP_Lamm_sf(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),Uua),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue))),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_Pro4465521767867336112on_nat(aa_VEB1870382382330055850on_nat(aa_arr666314619328815135on_nat(aa_nat8253036615571765984on_nat(aa_nat2001721438062390249on_nat(aTP_Lamm_se(Uu),Uua),Uub),Uuc),Uud),Uue))) ) ).

% ATP.lambda_821
tff(fact_10997_ATP_Olambda__822,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aa_nat8243565228001313504on_nat(aTP_Lamm_hc(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),Uua),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue))),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_Pro4465521767867336112on_nat(aa_VEB1870382382330055850on_nat(aa_arr666314619328815135on_nat(aa_nat8253036615571765984on_nat(aa_nat2001721438062390249on_nat(aTP_Lamm_hb(Uu),Uua),Uub),Uuc),Uud),Uue))) ) ).

% ATP.lambda_822
tff(fact_10998_ATP_Olambda__823,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aTP_Lamm_au(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uu),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),xa),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue))),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_Pro4465521767867336112on_nat(aa_VEB6043830584257997615on_nat(aa_lis5985300925819340672on_nat(aa_VEB1019093884839074554on_nat(aa_arr1617095457630768239on_nat(aTP_Lamm_at(Uu),Uua),Uub),Uuc),Uud),Uue))) ) ).

% ATP.lambda_823
tff(fact_10999_ATP_Olambda__824,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aa_nat8243565228001313504on_nat(aTP_Lamm_rx(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uue)),
          heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uue))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),Uua),heap_T3487192422709364219on_nat(none_nat),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_VEB3566417204837707778on_nat(aa_arr5230852142804057699on_nat(aa_nat6371187903253094776on_nat(aa_nat1535423446758346053on_nat(aTP_Lamm_rw(Uu),Uua),Uub),Uuc),Uud))) ) ) ).

% ATP.lambda_824
tff(fact_11000_ATP_Olambda__825,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aa_nat8243565228001313504on_nat(aTP_Lamm_hi(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uue)),
          heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uue))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),Uua),heap_T3487192422709364219on_nat(none_nat),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_VEB3566417204837707778on_nat(aa_arr5230852142804057699on_nat(aa_nat6371187903253094776on_nat(aa_nat1535423446758346053on_nat(aTP_Lamm_hh(Uu),Uua),Uub),Uuc),Uud))) ) ) ).

% ATP.lambda_825
tff(fact_11001_ATP_Olambda__826,axiom,
    ! [Uu: fun_Pr238758252084143515Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: nat,Uue: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_nat7415925511494575164Heap_o(aTP_Lamm_sv(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
        $ite(
          Uua = Uud,
          aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
          $ite(
            Uua = Uue,
            aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uud),
              aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),Uua),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_arr494317875797454629Heap_o(aa_nat857754347168699366Heap_o(aa_nat8612037029889215215Heap_o(aTP_Lamm_su(Uu),Uua),Uub),Uuc))) ) ) ) ) ).

% ATP.lambda_826
tff(fact_11002_ATP_Olambda__827,axiom,
    ! [Uu: fun_VE7134199479003659425Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: nat,Uue: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_nat7415925511494575164Heap_o(aTP_Lamm_oq(Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
        $ite(
          Uua = Uud,
          aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
          $ite(
            Uua = Uue,
            aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uud),
              aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),Uua),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_arr494317875797454629Heap_o(aa_nat857754347168699366Heap_o(aa_nat8612037029889215215Heap_o(aTP_Lamm_op(Uu),Uua),Uub),Uuc))) ) ) ) ) ).

% ATP.lambda_827
tff(fact_11003_ATP_Olambda__828,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_sg(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aa_nat8243565228001313504on_nat(aTP_Lamm_sf(Uu),Uua),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_828
tff(fact_11004_ATP_Olambda__829,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_ry(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aa_nat8243565228001313504on_nat(aTP_Lamm_rx(Uu),Uua),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_829
tff(fact_11005_ATP_Olambda__830,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_hj(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aa_nat8243565228001313504on_nat(aTP_Lamm_hi(Uu),Uua),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_830
tff(fact_11006_ATP_Olambda__831,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_hd(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB7519107058437837345on_nat(aa_arr2984472677043443222on_nat(aa_nat7372410979875760471on_nat(aa_nat8243565228001313504on_nat(aTP_Lamm_hc(Uu),Uua),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_831
tff(fact_11007_ATP_Olambda__832,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: product_unit] : aa_Pro3379321066291785715Heap_o(aa_arr4590259283924850188Heap_o(aa_nat2073207392357089185Heap_o(aa_opt2919655305572010731Heap_o(aa_nat8312540953585231916Heap_o(aTP_Lamm_np(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o1442776274061689234at_nat(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc3505292621261808240Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_opt8863613227715573503Heap_o(aa_nat8734498203866763412Heap_o(aTP_Lamm_no(Uu),Uua),Uub),Uuc),Uud)),Uub) ).

% ATP.lambda_832
tff(fact_11008_ATP_Olambda__833,axiom,
    ! [Uu: fun_Pr238758252084143515Heap_o,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aa_nat1227502447850154924Heap_o(aTP_Lamm_sw(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o1442776274061689234at_nat(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc3505292621261808240Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_nat7415925511494575164Heap_o(aTP_Lamm_sv(Uu),Uua),Uuc),Uud)),Uub) ).

% ATP.lambda_833
tff(fact_11009_ATP_Olambda__834,axiom,
    ! [Uu: fun_VE7134199479003659425Heap_o,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aa_nat1227502447850154924Heap_o(aTP_Lamm_or(Uu),Uua),Uub),Uuc),Uud),Uue) = case_o1442776274061689234at_nat(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc3505292621261808240Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_nat7415925511494575164Heap_o(aTP_Lamm_oq(Uu),Uua),Uuc),Uud)),Uub) ).

% ATP.lambda_834
tff(fact_11010_ATP_Olambda__835,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: code_integer,Uue: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aTP_Lamm_asj((Uu)),(Uua)),(Uub)),(Uuc)),Uud),(Uue)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aTP_Lamm_asi((Uu)),(Uua)),(Uub)),(Uuc)),(Uue)),code_bit_cut_integer(Uud)) ).

% ATP.lambda_835
tff(fact_11011_ATP_Olambda__836,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB7687887442520674464_VEBTi(aa_arr2561095334397224705_VEBTi(aa_nat8715118376710468630_VEBTi(aTP_Lamm_yo(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),Uud),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uud),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uu)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB4322370096674025769_VEBTi(aa_arr4526045539431534346_VEBTi(aa_nat9189989511310912415_VEBTi(aTP_Lamm_yn(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_836
tff(fact_11012_ATP_Olambda__837,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] :
      aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_nat3780430034849525329_VEBTi(aa_nat6565337644025228958_VEBTi(aTP_Lamm_wd(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uub),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uub),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua)),aa_VEB4833742232818206615_VEBTi(aa_arr6831033733615195768_VEBTi(aa_nat7652049589537018893_VEBTi(aTP_Lamm_wc(Uu),Uuc),Uud),Uue)) ).

% ATP.lambda_837
tff(fact_11013_ATP_Olambda__838,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat,Uue: nat] : aa_nat947733127081326238on_nat(aa_Pro4465521767867336112on_nat(aa_VEB1870382382330055850on_nat(aa_arr666314619328815135on_nat(aa_nat8253036615571765984on_nat(aTP_Lamm_bh(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB358289369652911795on_nat(aa_arr4041137963511513512on_nat(aa_nat1363181691939120105on_nat(aTP_Lamm_bg(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_838
tff(fact_11014_ATP_Olambda__839,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat] : aa_nat2910048432523430448Heap_o(aa_lis5083133406047025005Heap_o(aa_arr6185401592279617542Heap_o(aa_nat1466035401240590747Heap_o(aa_nat7251126122756508648Heap_o(aTP_Lamm_tf(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat5592791454563588861Heap_o(aa_lis455817436297872826Heap_o(aa_arr1217223296832173011Heap_o(aa_nat6014765364524370664Heap_o(aa_nat4979062449914608309Heap_o(aTP_Lamm_te(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_839
tff(fact_11015_ATP_Olambda__840,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat] : aa_nat2910048432523430448Heap_o(aa_lis5083133406047025005Heap_o(aa_arr6185401592279617542Heap_o(aa_nat1466035401240590747Heap_o(aa_nat7251126122756508648Heap_o(aTP_Lamm_oe(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat5592791454563588861Heap_o(aa_lis455817436297872826Heap_o(aa_arr1217223296832173011Heap_o(aa_nat6014765364524370664Heap_o(aa_nat4979062449914608309Heap_o(aTP_Lamm_od(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_840
tff(fact_11016_ATP_Olambda__841,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: product_unit] : aa_Pro3379321066291785715Heap_o(aa_lis4443831354813659076Heap_o(aa_arr8974760139355794305Heap_o(aa_nat808430574047188546Heap_o(aa_nat9059937328556229451Heap_o(aTP_Lamm_tg(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_lis5083133406047025005Heap_o(aa_arr6185401592279617542Heap_o(aa_nat1466035401240590747Heap_o(aa_nat7251126122756508648Heap_o(aTP_Lamm_tf(Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_841
tff(fact_11017_ATP_Olambda__842,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_arr5230852142804057699on_nat(aa_nat6371187903253094776on_nat(aa_nat1535423446758346053on_nat(aTP_Lamm_rw(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_VEB6424020099669959819on_nat(aa_arr7085280839797930604on_nat(aa_nat1161784880827100417on_nat(aTP_Lamm_rv(Uu),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_842
tff(fact_11018_ATP_Olambda__843,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: product_unit] : aa_Pro3379321066291785715Heap_o(aa_lis4443831354813659076Heap_o(aa_arr8974760139355794305Heap_o(aa_nat808430574047188546Heap_o(aa_nat9059937328556229451Heap_o(aTP_Lamm_of(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind_nat_o(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_lis5083133406047025005Heap_o(aa_arr6185401592279617542Heap_o(aa_nat1466035401240590747Heap_o(aa_nat7251126122756508648Heap_o(aTP_Lamm_oe(Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_843
tff(fact_11019_ATP_Olambda__844,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat] : aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_arr5230852142804057699on_nat(aa_nat6371187903253094776on_nat(aa_nat1535423446758346053on_nat(aTP_Lamm_hh(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_VEB6424020099669959819on_nat(aa_arr7085280839797930604on_nat(aa_nat1161784880827100417on_nat(aTP_Lamm_hg(Uu),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_844
tff(fact_11020_ATP_Olambda__845,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: product_prod_nat_nat,Uud: nat,Uue: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB358289369652911795on_nat(aa_arr4041137963511513512on_nat(aTP_Lamm_af(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),Uue),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB6384125456408274455on_nat(aa_arr4162587047475849484on_nat(aTP_Lamm_ae(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_845
tff(fact_11021_ATP_Olambda__846,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_VEB6424020099669959819on_nat(aa_arr7085280839797930604on_nat(aa_nat1161784880827100417on_nat(aTP_Lamm_rv(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uue),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB216942459957533935on_nat(aa_arr2558136924908760784on_nat(aa_nat402565781209047397on_nat(aTP_Lamm_ru(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_846
tff(fact_11022_ATP_Olambda__847,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_VEB6424020099669959819on_nat(aa_arr7085280839797930604on_nat(aa_nat1161784880827100417on_nat(aTP_Lamm_hg(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uue),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB216942459957533935on_nat(aa_arr2558136924908760784on_nat(aa_nat402565781209047397on_nat(aTP_Lamm_hf(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_847
tff(fact_11023_ATP_Olambda__848,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: array_VEBT_VEBTi,Uub: list_VEBT_VEBT,Uuc: nat,Uud: nat,Uue: product_unit] : aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aTP_Lamm_ta(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T3040810144269856602EBTi_o(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),Uuc),aa_nat1164146675577659234Heap_o(aa_nat3058146694327238959Heap_o(aa_lis6732064841504156908Heap_o(aTP_Lamm_sz(Uu),Uub),Uuc),Uud)) ).

% ATP.lambda_848
tff(fact_11024_ATP_Olambda__849,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: array_VEBT_VEBTi,Uub: list_VEBT_VEBT,Uuc: nat,Uud: nat,Uue: product_unit] : aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aTP_Lamm_nz(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T3040810144269856602EBTi_o(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),Uuc),aa_nat1164146675577659234Heap_o(aa_nat3058146694327238959Heap_o(aa_lis6732064841504156908Heap_o(aTP_Lamm_ny(Uu),Uub),Uuc),Uud)) ).

% ATP.lambda_849
tff(fact_11025_ATP_Olambda__850,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: nat,Uue: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_lis455817436297872826Heap_o(aa_arr1217223296832173011Heap_o(aa_nat6014765364524370664Heap_o(aTP_Lamm_ni(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T7577940988442287570unit_o(refine_Imp_assert(( ( Uue = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
          & ( Uud = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) )),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aa_nat1269258765096487380Heap_o(aTP_Lamm_nh(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_850
tff(fact_11026_ATP_Olambda__851,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBT,Uuc: product_prod_nat_nat,Uud: nat,Uue: option_nat] :
      aa_opt9163293676379396302on_nat(aa_nat9127621225095157975on_nat(aa_Pro5040713308973091561on_nat(aa_VEB8830394031746293224on_nat(aa_arr3423563577619635969on_nat(aTP_Lamm_al(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T5999496708990702694on_nat(refine_Imp_assert(( ( Uue = none_nat )
          = ( aa_nat_option_nat(vEBT_vebt_pred(Uub),Uud) = none_nat ) )),aa_opt5089197864053947520on_nat(aa_Pro5009687527940591364on_nat(aa_arr722815569820881053on_nat(aTP_Lamm_ak(Uu),Uua),Uuc),Uue)) ).

% ATP.lambda_851
tff(fact_11027_ATP_Olambda__852,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_eu(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T5999496708990702694on_nat(refine_Imp_assert(aa_VEBT_VEBT_o(vEBT_is_Node,Uu)),aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aTP_Lamm_et(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_852
tff(fact_11028_ATP_Olambda__853,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aTP_Lamm_cb(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T5999496708990702694on_nat(refine_Imp_assert(aa_VEBT_VEBT_o(vEBT_is_Node,Uu)),aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aTP_Lamm_ca(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_853
tff(fact_11029_ATP_Olambda__854,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aa_nat1227502447850154924Heap_o(aTP_Lamm_nq(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T7577940988442287570unit_o(refine_Imp_assert(aa_VEBT_VEBT_o(vEBT_is_Node,Uu)),aa_arr4590259283924850188Heap_o(aa_nat2073207392357089185Heap_o(aa_opt2919655305572010731Heap_o(aa_nat8312540953585231916Heap_o(aTP_Lamm_np(Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_854
tff(fact_11030_ATP_Olambda__855,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat,Uue: nat] : aa_nat5408703056993926195_VEBTi(aa_Pro3150526172571800645_VEBTi(aa_VEB4890588985164552255_VEBTi(aa_arr5592147402706875956_VEBTi(aa_nat5963232373128642933_VEBTi(aTP_Lamm_yp(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uud)),aa_nat4282640288315822012_VEBTi(aa_VEB7687887442520674464_VEBTi(aa_arr2561095334397224705_VEBTi(aa_nat8715118376710468630_VEBTi(aTP_Lamm_yo(Uu),Uua),Uub),Uuc),Uue)) ).

% ATP.lambda_855
tff(fact_11031_ATP_Olambda__856,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB216942459957533935on_nat(aa_arr2558136924908760784on_nat(aTP_Lamm_cl(Uu),Uua),Uub),Uuc),Uud),Uue) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Uue),aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_VEB4635808744058847656on_nat(aa_arr3352113509897533961on_nat(aTP_Lamm_ck(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_856
tff(fact_11032_ATP_Olambda__857,axiom,
    ! [Uu: fun_He8247729123535230979_VEBTi,Uua: fun_op5127434927081926045_VEBTi,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aa_fun4118786628394255868_VEBTi(aTP_Lamm_kk(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea2545666323642607868_VEBTi(Uu,aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_857
tff(fact_11033_ATP_Olambda__858,axiom,
    ! [Uu: fun_He1783186893022644846on_nat,Uua: fun_op5127434927081926045_VEBTi,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_fun7829374822385380327on_nat(aTP_Lamm_kg(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea6259128614175166183on_nat(Uu,aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_858
tff(fact_11034_ATP_Olambda__859,axiom,
    ! [Uu: fun_He4624006784692404214Heap_o,Uua: fun_op5127434927081926045_VEBTi,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aa_fun7400618842136392575Heap_o(aTP_Lamm_ki(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea7612170549154561063Heap_o(Uu,aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_859
tff(fact_11035_ATP_Olambda__860,axiom,
    ! [Uu: fun_He2623223239018661208_VEBTi,Uua: fun_op5783775701855621896on_nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aa_fun8651712798330174033_VEBTi(aTP_Lamm_jy(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea2832269215923769937_VEBTi(Uu,aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_860
tff(fact_11036_ATP_Olambda__861,axiom,
    ! [Uu: fun_He8640631688564546883on_nat,Uua: fun_op5783775701855621896on_nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_fun5534811901068199356on_nat(aTP_Lamm_ju(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea6597239020403563964on_nat(Uu,aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_861
tff(fact_11037_ATP_Olambda__862,axiom,
    ! [Uu: fun_He5988046180898962017Heap_o,Uua: fun_op5783775701855621896on_nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aa_fun836855539625404138Heap_o(aTP_Lamm_jw(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea662945835469240978Heap_o(Uu,aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_862
tff(fact_11038_ATP_Olambda__863,axiom,
    ! [Uu: fun_He6517945810935171576_VEBTi,Uua: fun_op1763656958002597022Heap_o,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aa_fun7082494938152799553_VEBTi(aTP_Lamm_ke(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea4465279032261640929_VEBTi(Uu,aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_863
tff(fact_11039_ATP_Olambda__864,axiom,
    ! [Uu: fun_He6140748070426431971on_nat,Uua: fun_op1763656958002597022Heap_o,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_fun878802566620225708on_nat(aTP_Lamm_ka(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea41712926108436044on_nat(Uu,aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_864
tff(fact_11040_ATP_Olambda__865,axiom,
    ! [Uu: fun_He6597989805181038017Heap_o,Uua: fun_op1763656958002597022Heap_o,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aa_fun2115832704338266106Heap_o(aTP_Lamm_kc(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Hea4824394052061459970Heap_o(Uu,aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_865
tff(fact_11041_ATP_Olambda__866,axiom,
    ! [Uu: fun_Pr3650993782763339253T_VEBT,Uua: fun_op8229906066235313459T_VEBT,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT] : aa_VEB8692087746923902260T_VEBT(aa_lis2966823273985285125T_VEBT(aa_nat6705503460730637082T_VEBT(aa_opt2902932101092860132T_VEBT(aa_fun7544404396881244796T_VEBT(aTP_Lamm_ii(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_Pro5421721056952420150T_VEBT(Uu,aa_VEB8692087746923902260T_VEBT(aa_lis2966823273985285125T_VEBT(aa_nat6705503460730637082T_VEBT(aa_opt2902932101092860132T_VEBT(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_866
tff(fact_11042_ATP_Olambda__867,axiom,
    ! [Uu: fun_Pr7471953115884902709VEBT_o,Uua: fun_op8229906066235313459T_VEBT,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_lis4184305612057247525VEBT_o(aa_nat8798449810042143974VEBT_o(aa_opt7072554309604550194VEBT_o(aa_fun5124765266653172784VEBT_o(aTP_Lamm_ik(Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> aa_Pro5475924895142249886VEBT_o(Uu,aa_VEB8692087746923902260T_VEBT(aa_lis2966823273985285125T_VEBT(aa_nat6705503460730637082T_VEBT(aa_opt2902932101092860132T_VEBT(Uua,Uub),Uuc),Uud),Uue)) ) ).

% ATP.lambda_867
tff(fact_11043_ATP_Olambda__868,axiom,
    ! [Uu: fun_o_3001263932818058983T_VEBT,Uua: fun_op541075054904862521VEBT_o,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT] : aa_VEB8692087746923902260T_VEBT(aa_lis2966823273985285125T_VEBT(aa_nat6705503460730637082T_VEBT(aa_opt2902932101092860132T_VEBT(aa_fun8362160726713724924T_VEBT(aTP_Lamm_im(Uu),Uua),Uub),Uuc),Uud),Uue) = aa_o_P8223745691486049304T_VEBT(Uu,aa_VEBT_VEBT_o(aa_lis4184305612057247525VEBT_o(aa_nat8798449810042143974VEBT_o(aa_opt7072554309604550194VEBT_o(Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_868
tff(fact_11044_ATP_Olambda__869,axiom,
    ! [Uu: fun_o_o,Uua: fun_op541075054904862521VEBT_o,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_lis4184305612057247525VEBT_o(aa_nat8798449810042143974VEBT_o(aa_opt7072554309604550194VEBT_o(aa_fun2455880738235528880VEBT_o(aTP_Lamm_io(Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> aa_o_o(Uu,aa_VEBT_VEBT_o(aa_lis4184305612057247525VEBT_o(aa_nat8798449810042143974VEBT_o(aa_opt7072554309604550194VEBT_o(Uua,Uub),Uuc),Uud),Uue)) ) ).

% ATP.lambda_869
tff(fact_11045_ATP_Olambda__870,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: option4927543243414619207at_nat] : aa_opt6266093870483638640Heap_o(aa_arr6198139058929667337Heap_o(aa_nat436397609992601886Heap_o(aa_opt4438414422311449448Heap_o(aa_nat1884043827438135593Heap_o(aTP_Lamm_tj(Uu),Uua),Uub),Uuc),Uud),Uue) = produc5946672270950774087Heap_o(aa_opt7844807959364121492Heap_o(aa_arr79663438428320465Heap_o(aa_nat2087714105643485586Heap_o(aa_opt9028207981337828190Heap_o(aa_nat2171064521259968883Heap_o(aTP_Lamm_ti(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_870
tff(fact_11046_ATP_Olambda__871,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: option4927543243414619207at_nat] : aa_opt6266093870483638640Heap_o(aa_arr6198139058929667337Heap_o(aa_nat436397609992601886Heap_o(aa_opt4438414422311449448Heap_o(aa_nat1884043827438135593Heap_o(aTP_Lamm_oi(Uu),Uua),Uub),Uuc),Uud),Uue) = produc5946672270950774087Heap_o(aa_opt7844807959364121492Heap_o(aa_arr79663438428320465Heap_o(aa_nat2087714105643485586Heap_o(aa_opt9028207981337828190Heap_o(aa_nat2171064521259968883Heap_o(aTP_Lamm_oh(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_871
tff(fact_11047_ATP_Olambda__872,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: option4927543243414619207at_nat,Uue: nat] : aa_nat6527088363260489928Heap_o(aa_opt7844807959364121492Heap_o(aa_arr79663438428320465Heap_o(aa_nat2087714105643485586Heap_o(aa_opt9028207981337828190Heap_o(aTP_Lamm_nm(Uu),Uua),Uub),Uuc),Uud),Uue) = produc5048428016959714504Heap_o(aa_nat3831830007592743808Heap_o(aa_opt3748239118003421002Heap_o(aa_arr581739785191926499Heap_o(aa_nat105892859602166008Heap_o(aa_opt8417070463701818562Heap_o(aTP_Lamm_nl(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_872
tff(fact_11048_ATP_Olambda__873,axiom,
    ! [Uu: option4927543243414619207at_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: option4927543243414619207at_nat,Uue: nat] : aa_nat2075221075791484188on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aTP_Lamm_ax(Uu),Uua),Uub),Uuc),Uud),Uue) = produc6124225815318652422on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aTP_Lamm_aw(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_873
tff(fact_11049_ATP_Olambda__874,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi] :
      aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aa_nat3009484633486537743_VEBTi(aa_VEB600817258366935214_VEBTi(aTP_Lamm_zr(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = case_o1356590567247012107at_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uud),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uuc,Uud,Uue),Uuf)),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uub),Uub)),Uud,Uue),Uuf))),aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aa_VEB7645014810616801796_VEBTi(aTP_Lamm_zq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uuc) ).

% ATP.lambda_874
tff(fact_11050_ATP_Olambda__875,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi] :
      aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(aa_nat3009484633486537743_VEBTi(aa_VEB600817258366935214_VEBTi(aTP_Lamm_xa(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = case_o1356590567247012107at_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uud),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uuc,Uud,Uue),Uuf)),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uub),Uub)),Uud,Uue),Uuf))),aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aa_VEB7645014810616801796_VEBTi(aTP_Lamm_wz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uuc) ).

% ATP.lambda_875
tff(fact_11051_ATP_Olambda__876,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: vEBT_VEBTi,Uuf: option_nat] :
      aa_opt9163293676379396302on_nat(aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_VEB4635808744058847656on_nat(aa_arr3352113509897533961on_nat(aTP_Lamm_ck(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(
        ( ( Uuf != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,Uuc)),Uuf) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_succi,Uue),Uuc),aa_nat9127621225095157975on_nat(aTP_Lamm_a(Uu),Uud)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_succi,Uub),Uud),aa_arr3757929795460847279on_nat(aTP_Lamm_cj(Uu),Uua)) ) ).

% ATP.lambda_876
tff(fact_11052_ATP_Olambda__877,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat] :
      aa_Pro1395359565518501564_VEBTi(aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aTP_Lamm_aap(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uuc),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uub,Uuc,Uud),Uue)),heap_T7982501707604696571_VEBTi(refine_Imp_assert(aa_VEBT_VEBT_o(vEBT_is_Node,Uu)),aa_Pro8034723660955798012_VEBTi(aa_VEB4262372667711301016_VEBTi(aa_arr9036764603281787897_VEBTi(aa_nat2771851207891374350_VEBTi(aa_opt7594342031245826264_VEBTi(aa_nat4475881176255417753_VEBTi(aTP_Lamm_aao(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf))) ).

% ATP.lambda_877
tff(fact_11053_ATP_Olambda__878,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat] :
      aa_Pro1395359565518501564_VEBTi(aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aTP_Lamm_yd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uuc),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uub,Uuc,Uud),Uue)),heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uuf)),aa_Pro3150526172571800645_VEBTi(aa_VEB4890588985164552255_VEBTi(aa_arr5592147402706875956_VEBTi(aa_nat5963232373128642933_VEBTi(aa_nat7459085003528029950_VEBTi(aTP_Lamm_yc(Uu),Uua),Uuc),Uud),Uue),Uuf))) ).

% ATP.lambda_878
tff(fact_11054_ATP_Olambda__879,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat] :
      aa_Pro1395359565518501564_VEBTi(aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aTP_Lamm_xo(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uuc),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uub,Uuc,Uud),Uue)),heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uuf)),aa_Pro3150526172571800645_VEBTi(aa_VEB4890588985164552255_VEBTi(aa_arr5592147402706875956_VEBTi(aa_nat5963232373128642933_VEBTi(aa_nat7459085003528029950_VEBTi(aTP_Lamm_xn(Uu),Uua),Uuc),Uud),Uue),Uuf))) ).

% ATP.lambda_879
tff(fact_11055_ATP_Olambda__880,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aTP_Lamm_bv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uuf)),Uu),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uuf))),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_Pro4465521767867336112on_nat(aa_VEB6043830584257997615on_nat(aa_lis5985300925819340672on_nat(aa_VEB1019093884839074554on_nat(aa_arr1617095457630768239on_nat(aa_nat5782952277816596784on_nat(aTP_Lamm_bu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf))) ) ).

% ATP.lambda_880
tff(fact_11056_ATP_Olambda__881,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aTP_Lamm_eo(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uuf)),
          heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uuf))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uuf)),Uu),heap_T3487192422709364219on_nat(none_nat),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_VEB1312864128696253885on_nat(aa_lis1761857624600698434on_nat(aa_VEB4172908723204034014on_nat(aa_arr2072991348567353407on_nat(aa_nat7185663501778404180on_nat(aTP_Lamm_en(Uu),Uua),Uub),Uuc),Uud),Uue))) ) ) ).

% ATP.lambda_881
tff(fact_11057_ATP_Olambda__882,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: nat,Uuf: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_opt8863613227715573503Heap_o(aa_nat8734498203866763412Heap_o(aTP_Lamm_no(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uuc),one_one_nat),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
        $ite(
          Uua = Uue,
          aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
          $ite(
            Uua = Uuf,
            aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uue),
              aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),Uua),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc1330493526443650053Heap_o(aa_arr6198139058929667337Heap_o(aa_nat436397609992601886Heap_o(aa_opt4438414422311449448Heap_o(aTP_Lamm_nn(Uua),Uub),Uuc),Uud),vEBT_c634343235235684882T_VEBT(aTP_Lamm_az,aTP_Lamm_ba,Uu))) ) ) ) ) ).

% ATP.lambda_882
tff(fact_11058_ATP_Olambda__883,axiom,
    ! [Uu: option4927543243414619207at_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aTP_Lamm_av(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aTP_Lamm_au(Uua),Uub),Uuc),Uud),Uue),Uu) ).

% ATP.lambda_883
tff(fact_11059_ATP_Olambda__884,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: product_unit] : aa_Pro3379321066291785715Heap_o(aa_arr4590259283924850188Heap_o(aa_nat2073207392357089185Heap_o(aa_opt2919655305572010731Heap_o(aa_nat8312540953585231916Heap_o(aa_VEB1825396708860521035Heap_o(aTP_Lamm_tl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = case_o1442776274061689234at_nat(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc3505292621261808240Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_opt8863613227715573503Heap_o(aa_nat8734498203866763412Heap_o(aa_VEB8906299275298315467Heap_o(aTP_Lamm_tk(Uu),Uua),Uub),Uuc),Uud),Uue)),Uuc) ).

% ATP.lambda_884
tff(fact_11060_ATP_Olambda__885,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: product_unit] : aa_Pro3379321066291785715Heap_o(aa_arr4590259283924850188Heap_o(aa_nat2073207392357089185Heap_o(aa_opt2919655305572010731Heap_o(aa_nat8312540953585231916Heap_o(aa_VEB1825396708860521035Heap_o(aTP_Lamm_ok(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = case_o1442776274061689234at_nat(aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc3505292621261808240Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_opt8863613227715573503Heap_o(aa_nat8734498203866763412Heap_o(aa_VEB8906299275298315467Heap_o(aTP_Lamm_oj(Uu),Uua),Uub),Uuc),Uud),Uue)),Uuc) ).

% ATP.lambda_885
tff(fact_11061_ATP_Olambda__886,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aTP_Lamm_et(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = produc2190226783763740553on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aTP_Lamm_es(Uu),Uua),Uub),Uuc),Uud),Uue),vEBT_c634343235235684882T_VEBT(aTP_Lamm_az,aTP_Lamm_ba,Uu)) ).

% ATP.lambda_886
tff(fact_11062_ATP_Olambda__887,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aTP_Lamm_ca(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = produc2190226783763740553on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aTP_Lamm_bz(Uu),Uua),Uub),Uuc),Uud),Uue),vEBT_c634343235235684882T_VEBT(aTP_Lamm_az,aTP_Lamm_ba,Uu)) ).

% ATP.lambda_887
tff(fact_11063_ATP_Olambda__888,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: code_integer,Uuf: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aTP_Lamm_asi((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),Uue),(Uuf)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aTP_Lamm_ash((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uuf)),code_bit_cut_integer(Uue)) ).

% ATP.lambda_888
tff(fact_11064_ATP_Olambda__889,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB4322370096674025769_VEBTi(aa_arr4526045539431534346_VEBTi(aa_nat9189989511310912415_VEBTi(aTP_Lamm_yn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),Uud),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uu),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uud)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB8340470449056147762_VEBTi(aa_arr7322700162901495443_VEBTi(aa_nat4321419670308649640_VEBTi(aTP_Lamm_ym(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_889
tff(fact_11065_ATP_Olambda__890,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB7687887442520674464_VEBTi(aa_arr2561095334397224705_VEBTi(aa_nat8715118376710468630_VEBTi(aa_nat5853910169739908579_VEBTi(aTP_Lamm_yb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uue),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uue),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB4322370096674025769_VEBTi(aa_arr4526045539431534346_VEBTi(aa_nat9189989511310912415_VEBTi(aa_nat4025588890810027244_VEBTi(aTP_Lamm_ya(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_890
tff(fact_11066_ATP_Olambda__891,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB7687887442520674464_VEBTi(aa_arr2561095334397224705_VEBTi(aa_nat8715118376710468630_VEBTi(aa_nat5853910169739908579_VEBTi(aTP_Lamm_xm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uue),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uue),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB4322370096674025769_VEBTi(aa_arr4526045539431534346_VEBTi(aa_nat9189989511310912415_VEBTi(aa_nat4025588890810027244_VEBTi(aTP_Lamm_xl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_891
tff(fact_11067_ATP_Olambda__892,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat] : aa_nat947733127081326238on_nat(aa_VEB1312864128696253885on_nat(aa_lis1761857624600698434on_nat(aa_VEB4172908723204034014on_nat(aa_arr2072991348567353407on_nat(aa_nat7185663501778404180on_nat(aTP_Lamm_en(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_VEB4460395522175602758on_nat(aa_lis6278591930053894219on_nat(aa_VEB6094637369740064103on_nat(aa_arr5396080262017029960on_nat(aa_nat7337942549949891037on_nat(aTP_Lamm_em(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_892
tff(fact_11068_ATP_Olambda__893,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat] : aa_nat947733127081326238on_nat(aa_Pro4465521767867336112on_nat(aa_VEB1870382382330055850on_nat(aa_arr666314619328815135on_nat(aa_nat8253036615571765984on_nat(aa_nat2001721438062390249on_nat(aTP_Lamm_se(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB358289369652911795on_nat(aa_arr4041137963511513512on_nat(aa_nat1363181691939120105on_nat(aa_nat4363128851979299954on_nat(aTP_Lamm_sd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_893
tff(fact_11069_ATP_Olambda__894,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat] : aa_nat947733127081326238on_nat(aa_Pro4465521767867336112on_nat(aa_VEB1870382382330055850on_nat(aa_arr666314619328815135on_nat(aa_nat8253036615571765984on_nat(aa_nat2001721438062390249on_nat(aTP_Lamm_hb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB358289369652911795on_nat(aa_arr4041137963511513512on_nat(aa_nat1363181691939120105on_nat(aa_nat4363128851979299954on_nat(aTP_Lamm_ha(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_894
tff(fact_11070_ATP_Olambda__895,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat,Uue: nat,Uuf: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB358289369652911795on_nat(aa_arr4041137963511513512on_nat(aa_nat1363181691939120105on_nat(aTP_Lamm_bg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uuf),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB6384125456408274455on_nat(aa_arr4162587047475849484on_nat(aa_nat5386638315231209293on_nat(aTP_Lamm_bf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_895
tff(fact_11071_ATP_Olambda__896,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_art(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uue),Uuf)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
        & ! [I4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uud))
           => ( ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,nth_VEBT_VEBT(Uua,I4)),X_12)
            <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Uub),I4) ) )
        & $ite(
            Uue = Uuf,
            ! [X: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X),set_VEBT_VEBT2(Uua))
             => ~ ? [X6: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X),X6) ),
            ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),Uuf)
            & ! [X: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
               => ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),X)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),X)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_896
tff(fact_11072_ATP_Olambda__897,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: product_prod_nat_nat,Uuf: nat] : aa_nat947733127081326238on_nat(aa_Pro4465521767867336112on_nat(aa_VEB6043830584257997615on_nat(aa_lis5985300925819340672on_nat(aa_VEB1019093884839074554on_nat(aa_arr1617095457630768239on_nat(aTP_Lamm_at(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB8035251902059208504on_nat(aa_lis5065134719111324425on_nat(aa_VEB1776269892753909763on_nat(aa_arr3452137061177157880on_nat(aTP_Lamm_as(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_897
tff(fact_11073_ATP_Olambda__898,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat,Uuf: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_lis455817436297872826Heap_o(aa_arr1217223296832173011Heap_o(aa_nat6014765364524370664Heap_o(aa_nat4979062449914608309Heap_o(aTP_Lamm_te(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T7577940988442287570unit_o(refine_Imp_assert(( ( Uuf = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
          & ( Uue = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) )),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aa_nat1269258765096487380Heap_o(aa_nat3499107050721502685Heap_o(aTP_Lamm_td(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_898
tff(fact_11074_ATP_Olambda__899,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat,Uuf: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_lis455817436297872826Heap_o(aa_arr1217223296832173011Heap_o(aa_nat6014765364524370664Heap_o(aa_nat4979062449914608309Heap_o(aTP_Lamm_od(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T7577940988442287570unit_o(refine_Imp_assert(( ( Uuf = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
          & ( Uue = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) )),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aa_nat1269258765096487380Heap_o(aa_nat3499107050721502685Heap_o(aTP_Lamm_oc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_899
tff(fact_11075_ATP_Olambda__900,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBT,Uud: product_prod_nat_nat,Uue: nat,Uuf: option_nat] :
      aa_opt9163293676379396302on_nat(aa_nat9127621225095157975on_nat(aa_Pro5040713308973091561on_nat(aa_VEB8830394031746293224on_nat(aa_arr3423563577619635969on_nat(aa_nat2635311902590537750on_nat(aTP_Lamm_bm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T5999496708990702694on_nat(refine_Imp_assert(( ( Uuf = none_nat )
          = ( aa_nat_option_nat(vEBT_vebt_pred(Uuc),Uue) = none_nat ) )),aa_opt5089197864053947520on_nat(aa_Pro5009687527940591364on_nat(aa_arr722815569820881053on_nat(aa_nat5945554870259702194on_nat(aTP_Lamm_bl(Uu),Uua),Uub),Uud),Uuf)) ).

% ATP.lambda_900
tff(fact_11076_ATP_Olambda__901,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: nat,Uue: nat,Uuf: nat] : aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_lis5270594155775668615Heap_o(aa_arr7588197057154790688Heap_o(aa_nat5373961893125994933Heap_o(aTP_Lamm_ng(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T7577940988442287570unit_o(refine_Imp_assert(Uuf = size_s6755466524823107622T_VEBT(Uuc)),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_nat5674868329695104454Heap_o(aa_lis4631090546836902751Heap_o(aa_arr3453073379018417564Heap_o(aa_nat4764294232906884829Heap_o(aTP_Lamm_nf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_901
tff(fact_11077_ATP_Olambda__902,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_px(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T5999496708990702694on_nat(refine_Imp_assert(aa_VEBT_VEBT_o(vEBT_is_Node,Uua)),aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aa_VEB4878072573979604067on_nat(aTP_Lamm_pw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_902
tff(fact_11078_ATP_Olambda__903,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_pg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T5999496708990702694on_nat(refine_Imp_assert(aa_VEBT_VEBT_o(vEBT_is_Node,Uua)),aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aa_VEB4878072573979604067on_nat(aTP_Lamm_pf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_903
tff(fact_11079_ATP_Olambda__904,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_gx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T5999496708990702694on_nat(refine_Imp_assert(aa_VEBT_VEBT_o(vEBT_is_Node,Uua)),aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aa_VEB4878072573979604067on_nat(aTP_Lamm_gw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_904
tff(fact_11080_ATP_Olambda__905,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(aa_nat3457676333800472698on_nat(aa_VEB6681700655871274393on_nat(aTP_Lamm_gi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T5999496708990702694on_nat(refine_Imp_assert(aa_VEBT_VEBT_o(vEBT_is_Node,Uua)),aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aa_VEB4878072573979604067on_nat(aTP_Lamm_gh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_905
tff(fact_11081_ATP_Olambda__906,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aa_nat1227502447850154924Heap_o(aa_VEB6636772557416003555Heap_o(aTP_Lamm_tm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T7577940988442287570unit_o(refine_Imp_assert(aa_VEBT_VEBT_o(vEBT_is_Node,Uua)),aa_arr4590259283924850188Heap_o(aa_nat2073207392357089185Heap_o(aa_opt2919655305572010731Heap_o(aa_nat8312540953585231916Heap_o(aa_VEB1825396708860521035Heap_o(aTP_Lamm_tl(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_906
tff(fact_11082_ATP_Olambda__907,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi] : aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(aa_nat1227502447850154924Heap_o(aa_VEB6636772557416003555Heap_o(aTP_Lamm_ol(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T7577940988442287570unit_o(refine_Imp_assert(aa_VEBT_VEBT_o(vEBT_is_Node,Uua)),aa_arr4590259283924850188Heap_o(aa_nat2073207392357089185Heap_o(aa_opt2919655305572010731Heap_o(aa_nat8312540953585231916Heap_o(aa_VEB1825396708860521035Heap_o(aTP_Lamm_ok(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_907
tff(fact_11083_ATP_Olambda__908,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat] : aa_nat5408703056993926195_VEBTi(aa_Pro3150526172571800645_VEBTi(aa_VEB4890588985164552255_VEBTi(aa_arr5592147402706875956_VEBTi(aa_nat5963232373128642933_VEBTi(aa_nat7459085003528029950_VEBTi(aTP_Lamm_yc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),aa_nat4282640288315822012_VEBTi(aa_VEB7687887442520674464_VEBTi(aa_arr2561095334397224705_VEBTi(aa_nat8715118376710468630_VEBTi(aa_nat5853910169739908579_VEBTi(aTP_Lamm_yb(Uu),Uua),Uub),Uuc),Uud),Uuf)) ).

% ATP.lambda_908
tff(fact_11084_ATP_Olambda__909,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat] : aa_nat5408703056993926195_VEBTi(aa_Pro3150526172571800645_VEBTi(aa_VEB4890588985164552255_VEBTi(aa_arr5592147402706875956_VEBTi(aa_nat5963232373128642933_VEBTi(aa_nat7459085003528029950_VEBTi(aTP_Lamm_xn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),aa_nat4282640288315822012_VEBTi(aa_VEB7687887442520674464_VEBTi(aa_arr2561095334397224705_VEBTi(aa_nat8715118376710468630_VEBTi(aa_nat5853910169739908579_VEBTi(aTP_Lamm_xm(Uu),Uua),Uub),Uuc),Uud),Uuf)) ).

% ATP.lambda_909
tff(fact_11085_ATP_Olambda__910,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: product_prod_nat_nat,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB6384125456408274455on_nat(aa_arr4162587047475849484on_nat(aTP_Lamm_ae(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uuf),aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_Pro2771509044941117526on_nat(aa_VEB8816156001103073488on_nat(aa_arr6004237943432320069on_nat(aTP_Lamm_ad(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_910
tff(fact_11086_ATP_Olambda__911,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB216942459957533935on_nat(aa_arr2558136924908760784on_nat(aa_nat402565781209047397on_nat(aTP_Lamm_ru(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Uuf),aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_VEB4635808744058847656on_nat(aa_arr3352113509897533961on_nat(aa_nat8985933306058800414on_nat(aTP_Lamm_rt(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_911
tff(fact_11087_ATP_Olambda__912,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB216942459957533935on_nat(aa_arr2558136924908760784on_nat(aa_nat402565781209047397on_nat(aTP_Lamm_hf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Uuf),aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_VEB4635808744058847656on_nat(aa_arr3352113509897533961on_nat(aa_nat8985933306058800414on_nat(aTP_Lamm_he(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_912
tff(fact_11088_ATP_Olambda__913,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: nat,Uue: nat,Uuf: product_unit] : aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aa_nat1269258765096487380Heap_o(aTP_Lamm_nh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind_nat_o(array_len_VEBT_VEBTi(Uub),aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_lis5270594155775668615Heap_o(aa_arr7588197057154790688Heap_o(aa_nat5373961893125994933Heap_o(aTP_Lamm_ng(Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_913
tff(fact_11089_ATP_Olambda__914,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: option4927543243414619207at_nat] : aa_opt8585651768545592073_VEBTi(aa_Pro4062752345418257877_VEBTi(aa_VEB8867863569893255025_VEBTi(aa_arr3828982135699130450_VEBTi(aa_nat5629247741253790183_VEBTi(aa_opt9205748023383517873_VEBTi(aTP_Lamm_aan(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = produc7864503674192730076_VEBTi(aa_opt7507675525305149691_VEBTi(aa_Pro1234189438217840325_VEBTi(aa_VEB660175962507042495_VEBTi(aa_arr8367260621226259636_VEBTi(aa_nat5784496157376858101_VEBTi(aa_opt81624670976341313_VEBTi(aTP_Lamm_aam(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_914
tff(fact_11090_ATP_Olambda__915,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: option4927543243414619207at_nat,Uuf: nat] : aa_nat6527088363260489928Heap_o(aa_opt7844807959364121492Heap_o(aa_arr79663438428320465Heap_o(aa_nat2087714105643485586Heap_o(aa_opt9028207981337828190Heap_o(aa_nat2171064521259968883Heap_o(aTP_Lamm_ti(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = produc5048428016959714504Heap_o(aa_nat3831830007592743808Heap_o(aa_opt3748239118003421002Heap_o(aa_arr581739785191926499Heap_o(aa_nat105892859602166008Heap_o(aa_opt8417070463701818562Heap_o(aa_nat8831994779944892035Heap_o(aTP_Lamm_th(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_915
tff(fact_11091_ATP_Olambda__916,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: option4927543243414619207at_nat,Uuf: nat] : aa_nat6527088363260489928Heap_o(aa_opt7844807959364121492Heap_o(aa_arr79663438428320465Heap_o(aa_nat2087714105643485586Heap_o(aa_opt9028207981337828190Heap_o(aa_nat2171064521259968883Heap_o(aTP_Lamm_oh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = produc5048428016959714504Heap_o(aa_nat3831830007592743808Heap_o(aa_opt3748239118003421002Heap_o(aa_arr581739785191926499Heap_o(aa_nat105892859602166008Heap_o(aa_opt8417070463701818562Heap_o(aa_nat8831994779944892035Heap_o(aTP_Lamm_og(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_916
tff(fact_11092_ATP_Olambda__917,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: option4927543243414619207at_nat] : aa_opt1657869255803490932on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aTP_Lamm_es(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = produc2606485630176857543on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aTP_Lamm_er(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_917
tff(fact_11093_ATP_Olambda__918,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: option4927543243414619207at_nat] : aa_opt1657869255803490932on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aTP_Lamm_bz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = produc2606485630176857543on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aTP_Lamm_by(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_918
tff(fact_11094_ATP_Olambda__919,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: code_integer] : aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aa_o_f3038258172971468011o_char(aTP_Lamm_asg((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),Uug) = char2((Uu),(Uua),(Uub),(Uuc),(Uud),(Uue),(Uuf)) ).

% ATP.lambda_919
tff(fact_11095_ATP_Olambda__920,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi,Uug: option_nat] :
      aa_opt9163293676379396302on_nat(aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_VEB4635808744058847656on_nat(aa_arr3352113509897533961on_nat(aa_nat8985933306058800414on_nat(aTP_Lamm_rt(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        ( ( Uug != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,Uud)),Uug) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(produc1489253303066280154on_nat(Uu),Uuf),Uud),aa_nat9127621225095157975on_nat(aTP_Lamm_a(Uua),Uue)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(produc1489253303066280154on_nat(Uu),Uuc),Uue),aa_arr3757929795460847279on_nat(aTP_Lamm_cj(Uua),Uub)) ) ).

% ATP.lambda_920
tff(fact_11096_ATP_Olambda__921,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi,Uug: option_nat] :
      aa_opt9163293676379396302on_nat(aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_VEB4635808744058847656on_nat(aa_arr3352113509897533961on_nat(aa_nat8985933306058800414on_nat(aTP_Lamm_he(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        ( ( Uug != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,Uud)),Uug) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(Uu,Uuf),Uud),aa_nat9127621225095157975on_nat(aTP_Lamm_a(Uua),Uue)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(Uu,Uuc),Uue),aa_arr3757929795460847279on_nat(aTP_Lamm_cj(Uua),Uub)) ) ).

% ATP.lambda_921
tff(fact_11097_ATP_Olambda__922,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: product_prod_nat_nat,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi,Uug: option_nat] :
      aa_opt9163293676379396302on_nat(aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_Pro2771509044941117526on_nat(aa_VEB8816156001103073488on_nat(aa_arr6004237943432320069on_nat(aTP_Lamm_ad(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        ( ( Uug != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Uud)),Uug) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_predi,Uuf),Uud),aa_nat9127621225095157975on_nat(aTP_Lamm_a(Uu),Uue)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_predi,Uub),Uue),aa_Pro4885747571561342816on_nat(aa_arr7927785801258559901on_nat(aTP_Lamm_ac(Uu),Uua),Uuc)) ) ).

% ATP.lambda_922
tff(fact_11098_ATP_Olambda__923,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: nat,Uue: nat,Uuf: nat,Uug: product_unit] :
      aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_nat5674868329695104454Heap_o(aa_lis4631090546836902751Heap_o(aa_arr3453073379018417564Heap_o(aa_nat4764294232906884829Heap_o(aTP_Lamm_nf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uud),Uuf),
        heap_T7577940988442287570unit_o(refine_Imp_assert(( ( Uud = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uud),size_s6755466524823107622T_VEBT(Uuc)) )),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aTP_Lamm_ne(Uub),Uuc),Uud),Uue)),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) ) ).

% ATP.lambda_923
tff(fact_11099_ATP_Olambda__924,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: product_prod_nat_nat] :
      aa_Pro1395359565518501564_VEBTi(aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aa_VEB7645014810616801796_VEBTi(aTP_Lamm_zq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uud),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uuc,Uud,Uue),Uuf)),heap_T7982501707604696571_VEBTi(refine_Imp_assert(aa_VEBT_VEBT_o(vEBT_is_Node,Uua)),aa_Pro8034723660955798012_VEBTi(aa_VEB4262372667711301016_VEBTi(aa_arr9036764603281787897_VEBTi(aa_nat2771851207891374350_VEBTi(aa_opt7594342031245826264_VEBTi(aa_nat4475881176255417753_VEBTi(aa_VEB8590999954632166712_VEBTi(aTP_Lamm_zp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))) ).

% ATP.lambda_924
tff(fact_11100_ATP_Olambda__925,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: product_prod_nat_nat] :
      aa_Pro1395359565518501564_VEBTi(aa_VEB6393524116121717046_VEBTi(aa_arr8059685486658508715_VEBTi(aa_nat7594302058568047468_VEBTi(aa_opt7720280829887018808_VEBTi(aa_nat6605875721633857101_VEBTi(aa_VEB7645014810616801796_VEBTi(aTP_Lamm_wz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uud),one_one_nat),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Uuc,Uud,Uue),Uuf)),heap_T7982501707604696571_VEBTi(refine_Imp_assert(aa_VEBT_VEBT_o(vEBT_is_Node,Uua)),aa_Pro8034723660955798012_VEBTi(aa_VEB4262372667711301016_VEBTi(aa_arr9036764603281787897_VEBTi(aa_nat2771851207891374350_VEBTi(aa_opt7594342031245826264_VEBTi(aa_nat4475881176255417753_VEBTi(aa_VEB8590999954632166712_VEBTi(aTP_Lamm_wy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))) ).

% ATP.lambda_925
tff(fact_11101_ATP_Olambda__926,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aa_nat1315775135507255344on_nat(aTP_Lamm_pr(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uug)),Uua),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uug))),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_Pro4465521767867336112on_nat(aa_VEB6043830584257997615on_nat(aa_lis5985300925819340672on_nat(aa_VEB1019093884839074554on_nat(aa_arr1617095457630768239on_nat(aa_nat5782952277816596784on_nat(aa_nat7314605301869835321on_nat(aTP_Lamm_pq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))) ) ).

% ATP.lambda_926
tff(fact_11102_ATP_Olambda__927,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aa_nat1315775135507255344on_nat(aTP_Lamm_gc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uug)),Uua),heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uug))),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_Pro4465521767867336112on_nat(aa_VEB6043830584257997615on_nat(aa_lis5985300925819340672on_nat(aa_VEB1019093884839074554on_nat(aa_arr1617095457630768239on_nat(aa_nat5782952277816596784on_nat(aa_nat7314605301869835321on_nat(aTP_Lamm_gb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))) ) ).

% ATP.lambda_927
tff(fact_11103_ATP_Olambda__928,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aa_nat1315775135507255344on_nat(aTP_Lamm_pa(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uug)),
          heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uug))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uug)),Uua),heap_T3487192422709364219on_nat(none_nat),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_VEB1312864128696253885on_nat(aa_lis1761857624600698434on_nat(aa_VEB4172908723204034014on_nat(aa_arr2072991348567353407on_nat(aa_nat7185663501778404180on_nat(aa_nat4144773949132746529on_nat(aTP_Lamm_oz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf))) ) ) ).

% ATP.lambda_928
tff(fact_11104_ATP_Olambda__929,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat] :
      aa_Pro7675874045790911655on_nat(aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aa_nat1315775135507255344on_nat(aTP_Lamm_gr(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uub),one_one_nat),
        heap_T3487192422709364219on_nat(none_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uug)),
          heap_T3487192422709364219on_nat(aa_nat_option_nat(some_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uug))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uug)),Uua),heap_T3487192422709364219on_nat(none_nat),heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_VEB1312864128696253885on_nat(aa_lis1761857624600698434on_nat(aa_VEB4172908723204034014on_nat(aa_arr2072991348567353407on_nat(aa_nat7185663501778404180on_nat(aa_nat4144773949132746529on_nat(aTP_Lamm_gq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf))) ) ) ).

% ATP.lambda_929
tff(fact_11105_ATP_Olambda__930,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: nat,Uug: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_opt8863613227715573503Heap_o(aa_nat8734498203866763412Heap_o(aa_VEB8906299275298315467Heap_o(aTP_Lamm_tk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uud),one_one_nat),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
        $ite(
          Uub = Uuf,
          aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
          $ite(
            Uub = Uug,
            aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uuf),
              aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uug),Uub),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc1330493526443650053Heap_o(aa_arr6198139058929667337Heap_o(aa_nat436397609992601886Heap_o(aa_opt4438414422311449448Heap_o(aa_nat1884043827438135593Heap_o(aTP_Lamm_tj(Uu),Uub),Uuc),Uud),Uue),vEBT_c634343235235684882T_VEBT(aTP_Lamm_az,aTP_Lamm_ba,Uua))) ) ) ) ) ).

% ATP.lambda_930
tff(fact_11106_ATP_Olambda__931,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: nat,Uug: nat] :
      aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_arr9046104431811898738Heap_o(aa_nat272996766647252915Heap_o(aa_opt8863613227715573503Heap_o(aa_nat8734498203866763412Heap_o(aa_VEB8906299275298315467Heap_o(aTP_Lamm_oj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uud),one_one_nat),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
        $ite(
          Uub = Uuf,
          aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
          $ite(
            Uub = Uug,
            aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$true),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uuf),
              aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uug),Uub),aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false),produc1330493526443650053Heap_o(aa_arr6198139058929667337Heap_o(aa_nat436397609992601886Heap_o(aa_opt4438414422311449448Heap_o(aa_nat1884043827438135593Heap_o(aTP_Lamm_oi(Uu),Uub),Uuc),Uud),Uue),vEBT_c634343235235684882T_VEBT(aTP_Lamm_az,aTP_Lamm_ba,Uua))) ) ) ) ) ).

% ATP.lambda_931
tff(fact_11107_ATP_Olambda__932,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aTP_Lamm_ep(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aTP_Lamm_eo(Uu),Uub),Uuc),Uud),Uue),Uuf),Uua) ).

% ATP.lambda_932
tff(fact_11108_ATP_Olambda__933,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aTP_Lamm_bw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aTP_Lamm_bv(Uu),Uub),Uuc),Uud),Uue),Uuf),Uua) ).

% ATP.lambda_933
tff(fact_11109_ATP_Olambda__934,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aa_VEB4878072573979604067on_nat(aTP_Lamm_pw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2190226783763740553on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aa_VEB8598360222073186748on_nat(aTP_Lamm_pv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),vEBT_c634343235235684882T_VEBT(aTP_Lamm_az,aTP_Lamm_ba,Uua)) ).

% ATP.lambda_934
tff(fact_11110_ATP_Olambda__935,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aa_VEB4878072573979604067on_nat(aTP_Lamm_pf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2190226783763740553on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aa_VEB8598360222073186748on_nat(aTP_Lamm_pe(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),vEBT_c634343235235684882T_VEBT(aTP_Lamm_az,aTP_Lamm_ba,Uua)) ).

% ATP.lambda_935
tff(fact_11111_ATP_Olambda__936,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aa_VEB4878072573979604067on_nat(aTP_Lamm_gw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2190226783763740553on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aa_VEB8598360222073186748on_nat(aTP_Lamm_gv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),vEBT_c634343235235684882T_VEBT(aTP_Lamm_az,aTP_Lamm_ba,Uua)) ).

% ATP.lambda_936
tff(fact_11112_ATP_Olambda__937,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB7937681756907646997on_nat(aa_arr5896957451705369866on_nat(aa_nat3269078897214415691on_nat(aa_opt5459845657693669015on_nat(aa_nat8168140587832489004on_nat(aa_VEB4878072573979604067on_nat(aTP_Lamm_gh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2190226783763740553on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aa_VEB8598360222073186748on_nat(aTP_Lamm_gg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),vEBT_c634343235235684882T_VEBT(aTP_Lamm_az,aTP_Lamm_ba,Uua)) ).

% ATP.lambda_937
tff(fact_11113_ATP_Olambda__938,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: code_integer,Uug: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aTP_Lamm_ash((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),Uuf),(Uug)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aa_o_f3038258172971468011o_char(aTP_Lamm_asg((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uug)),code_bit_cut_integer(Uuf)) ).

% ATP.lambda_938
tff(fact_11114_ATP_Olambda__939,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat,Uug: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_Pro8034723660955798012_VEBTi(aa_VEB4262372667711301016_VEBTi(aa_arr9036764603281787897_VEBTi(aa_nat2771851207891374350_VEBTi(aa_opt7594342031245826264_VEBTi(aa_nat4475881176255417753_VEBTi(aTP_Lamm_aao(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2624314226134418078_VEBTi(aa_Pro4062752345418257877_VEBTi(aa_VEB8867863569893255025_VEBTi(aa_arr3828982135699130450_VEBTi(aa_nat5629247741253790183_VEBTi(aa_opt9205748023383517873_VEBTi(aTP_Lamm_aan(Uua),Uub),Uuc),Uud),Uue),Uuf),vEBT_c634343235235684882T_VEBT(aTP_Lamm_az,aTP_Lamm_ba,Uu)) ).

% ATP.lambda_939
tff(fact_11115_ATP_Olambda__940,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB4322370096674025769_VEBTi(aa_arr4526045539431534346_VEBTi(aa_nat9189989511310912415_VEBTi(aa_nat4025588890810027244_VEBTi(aTP_Lamm_ya(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uue),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uue)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB8340470449056147762_VEBTi(aa_arr7322700162901495443_VEBTi(aa_nat4321419670308649640_VEBTi(aa_nat695344571205917557_VEBTi(aTP_Lamm_xz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_940
tff(fact_11116_ATP_Olambda__941,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB4322370096674025769_VEBTi(aa_arr4526045539431534346_VEBTi(aa_nat9189989511310912415_VEBTi(aa_nat4025588890810027244_VEBTi(aTP_Lamm_xl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uue),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uue)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB8340470449056147762_VEBTi(aa_arr7322700162901495443_VEBTi(aa_nat4321419670308649640_VEBTi(aa_nat695344571205917557_VEBTi(aTP_Lamm_xk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_941
tff(fact_11117_ATP_Olambda__942,axiom,
    ! [Uu: nat,Uua: vEBT_VEBTi,Uub: nat,Uuc: nat,Uud: nat,Uue: nat,Uuf: $o,Uug: array_VEBT_VEBTi] :
      aa_arr4839786837114051083_VEBTi(aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB8654527159822295598_VEBTi(aTP_Lamm_yf(Uu),Uua),Uub),Uuc),Uud),Uue),(Uuf)),Uug) = heap_T1006145433769338483_VEBTi($ite((Uuf),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(vEBT_vebt_inserti,Uua),Uue),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Uua)),aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_nat3780430034849525329_VEBTi(aa_nat6565337644025228958_VEBTi(aTP_Lamm_wd(Uu),Uub),Uuc),Uud),Uug)) ).

% ATP.lambda_942
tff(fact_11118_ATP_Olambda__943,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat] : aa_nat947733127081326238on_nat(aa_Pro4465521767867336112on_nat(aa_VEB6043830584257997615on_nat(aa_lis5985300925819340672on_nat(aa_VEB1019093884839074554on_nat(aa_arr1617095457630768239on_nat(aa_nat5782952277816596784on_nat(aTP_Lamm_bu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB8035251902059208504on_nat(aa_lis5065134719111324425on_nat(aa_VEB1776269892753909763on_nat(aa_arr3452137061177157880on_nat(aa_nat8770382747262106937on_nat(aTP_Lamm_bt(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_943
tff(fact_11119_ATP_Olambda__944,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat] : aa_nat947733127081326238on_nat(aa_VEB1312864128696253885on_nat(aa_lis1761857624600698434on_nat(aa_VEB4172908723204034014on_nat(aa_arr2072991348567353407on_nat(aa_nat7185663501778404180on_nat(aa_nat4144773949132746529on_nat(aTP_Lamm_oz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_VEB4460395522175602758on_nat(aa_lis6278591930053894219on_nat(aa_VEB6094637369740064103on_nat(aa_arr5396080262017029960on_nat(aa_nat7337942549949891037on_nat(aa_nat8478446459054042410on_nat(aTP_Lamm_oy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_944
tff(fact_11120_ATP_Olambda__945,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat] : aa_nat947733127081326238on_nat(aa_VEB1312864128696253885on_nat(aa_lis1761857624600698434on_nat(aa_VEB4172908723204034014on_nat(aa_arr2072991348567353407on_nat(aa_nat7185663501778404180on_nat(aa_nat4144773949132746529on_nat(aTP_Lamm_gq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_VEB4460395522175602758on_nat(aa_lis6278591930053894219on_nat(aa_VEB6094637369740064103on_nat(aa_arr5396080262017029960on_nat(aa_nat7337942549949891037on_nat(aa_nat8478446459054042410on_nat(aTP_Lamm_gp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_945
tff(fact_11121_ATP_Olambda__946,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB8340470449056147762_VEBTi(aa_arr7322700162901495443_VEBTi(aa_nat4321419670308649640_VEBTi(aTP_Lamm_ym(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uuf),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB8966970847287376059_VEBTi(aa_arr4873757251330958748_VEBTi(aa_nat4224949603525894449_VEBTi(aTP_Lamm_yl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_946
tff(fact_11122_ATP_Olambda__947,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB358289369652911795on_nat(aa_arr4041137963511513512on_nat(aa_nat1363181691939120105on_nat(aa_nat4363128851979299954on_nat(aTP_Lamm_sd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uuc),Uug),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB6384125456408274455on_nat(aa_arr4162587047475849484on_nat(aa_nat5386638315231209293on_nat(aa_nat7318487011899192278on_nat(aTP_Lamm_sc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_947
tff(fact_11123_ATP_Olambda__948,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB358289369652911795on_nat(aa_arr4041137963511513512on_nat(aa_nat1363181691939120105on_nat(aa_nat4363128851979299954on_nat(aTP_Lamm_ha(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uuc),Uug),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB6384125456408274455on_nat(aa_arr4162587047475849484on_nat(aa_nat5386638315231209293on_nat(aa_nat7318487011899192278on_nat(aTP_Lamm_gz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_948
tff(fact_11124_ATP_Olambda__949,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aTP_Lamm_ej(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),Uuf),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB6116742807553843114on_nat(aa_lis2576702180724645807on_nat(aa_VEB3484668625599252427on_nat(aa_arr4903404720305071532on_nat(aTP_Lamm_ei(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_949
tff(fact_11125_ATP_Olambda__950,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi] :
      aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB6116742807553843114on_nat(aa_lis2576702180724645807on_nat(aa_VEB3484668625599252427on_nat(aa_arr4903404720305071532on_nat(aTP_Lamm_ei(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= nth_VEBT_VEBT(Uuc,Uuf),
        heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Uug),aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_VEB3020412023619788856on_nat(aa_VEB2867584020876442580on_nat(aa_arr7250883597209086773on_nat(aTP_Lamm_eh(Uu),Uua),Uub),Uud),Uue),Uuf),Uug),aktnode)) ) ).

% ATP.lambda_950
tff(fact_11126_ATP_Olambda__951,axiom,
    ! [Uu: option4927543243414619207at_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: option4927543243414619207at_nat,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT] :
      aa_VEB3253283475358696116on_nat(aa_lis6893462050766468025on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aTP_Lamm_aw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T5999496708990702694on_nat(refine_Imp_assert(( ( Uud = Uu )
          & ( Uue = Uua )
          & aa_VEBT_VEBT_o(vEBT_is_Node,ta) )),aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aTP_Lamm_av(Uu),Uua),Uub),Uuc),Uuf),Uug)) ).

% ATP.lambda_951
tff(fact_11127_ATP_Olambda__952,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: option4927543243414619207at_nat,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT] :
      aa_VEB8530157447395038106Heap_o(aa_lis4690688361886208363Heap_o(aa_nat3831830007592743808Heap_o(aa_opt3748239118003421002Heap_o(aa_arr581739785191926499Heap_o(aa_nat105892859602166008Heap_o(aa_opt8417070463701818562Heap_o(aTP_Lamm_nl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T7577940988442287570unit_o(refine_Imp_assert(( ( Uua = Uud )
          & ( Uub = Uue ) )),aa_lis4443831354813659076Heap_o(aa_arr8974760139355794305Heap_o(aa_nat808430574047188546Heap_o(aTP_Lamm_nk(Uu),Uub),Uuc),Uuf)) ).

% ATP.lambda_952
tff(fact_11128_ATP_Olambda__953,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_VEB4460395522175602758on_nat(aa_lis6278591930053894219on_nat(aa_VEB6094637369740064103on_nat(aa_arr5396080262017029960on_nat(aa_nat7337942549949891037on_nat(aTP_Lamm_em(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uuf = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_el(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_953
tff(fact_11129_ATP_Olambda__954,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB8035251902059208504on_nat(aa_lis5065134719111324425on_nat(aa_VEB1776269892753909763on_nat(aa_arr3452137061177157880on_nat(aTP_Lamm_as(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uuf = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aTP_Lamm_ar(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_954
tff(fact_11130_ATP_Olambda__955,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat] : aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_lis5270594155775668615Heap_o(aa_arr7588197057154790688Heap_o(aa_nat5373961893125994933Heap_o(aa_nat5062659879350109954Heap_o(aTP_Lamm_tc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T7577940988442287570unit_o(refine_Imp_assert(Uug = size_s6755466524823107622T_VEBT(Uud)),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_nat5674868329695104454Heap_o(aa_lis4631090546836902751Heap_o(aa_arr3453073379018417564Heap_o(aa_nat4764294232906884829Heap_o(aa_nat334188166746217062Heap_o(aTP_Lamm_tb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_955
tff(fact_11131_ATP_Olambda__956,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat] : aa_nat2910048432523430448Heap_o(aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_lis5270594155775668615Heap_o(aa_arr7588197057154790688Heap_o(aa_nat5373961893125994933Heap_o(aa_nat5062659879350109954Heap_o(aTP_Lamm_ob(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T7577940988442287570unit_o(refine_Imp_assert(Uug = size_s6755466524823107622T_VEBT(Uud)),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_nat5674868329695104454Heap_o(aa_lis4631090546836902751Heap_o(aa_arr3453073379018417564Heap_o(aa_nat4764294232906884829Heap_o(aa_nat334188166746217062Heap_o(aTP_Lamm_oa(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_956
tff(fact_11132_ATP_Olambda__957,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aTP_Lamm_ek(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T5999496708990702694on_nat(refine_Imp_assert(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),size_s6755466524823107622T_VEBT(Uuc))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aTP_Lamm_ej(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_957
tff(fact_11133_ATP_Olambda__958,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB6384125456408274455on_nat(aa_arr4162587047475849484on_nat(aa_nat5386638315231209293on_nat(aTP_Lamm_bf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uug),aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_Pro2771509044941117526on_nat(aa_VEB8816156001103073488on_nat(aa_arr6004237943432320069on_nat(aa_nat4340965201858767110on_nat(aTP_Lamm_be(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_958
tff(fact_11134_ATP_Olambda__959,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat,Uuf: nat,Uug: product_unit] : aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aa_nat1269258765096487380Heap_o(aa_nat3499107050721502685Heap_o(aTP_Lamm_td(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind_nat_o(array_len_VEBT_VEBTi(Uuc),aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_lis5270594155775668615Heap_o(aa_arr7588197057154790688Heap_o(aa_nat5373961893125994933Heap_o(aa_nat5062659879350109954Heap_o(aTP_Lamm_tc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_959
tff(fact_11135_ATP_Olambda__960,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat,Uuf: nat,Uug: product_unit] : aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aa_nat1269258765096487380Heap_o(aa_nat3499107050721502685Heap_o(aTP_Lamm_oc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind_nat_o(array_len_VEBT_VEBTi(Uuc),aa_nat5592791454563588861Heap_o(aa_nat7429013343984486730Heap_o(aa_lis5270594155775668615Heap_o(aa_arr7588197057154790688Heap_o(aa_nat5373961893125994933Heap_o(aa_nat5062659879350109954Heap_o(aTP_Lamm_ob(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_960
tff(fact_11136_ATP_Olambda__961,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: option4927543243414619207at_nat,Uug: nat] : aa_nat7217600295869438385_VEBTi(aa_opt7507675525305149691_VEBTi(aa_Pro1234189438217840325_VEBTi(aa_VEB660175962507042495_VEBTi(aa_arr8367260621226259636_VEBTi(aa_nat5784496157376858101_VEBTi(aa_opt81624670976341313_VEBTi(aTP_Lamm_aam(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc9050507437146595227_VEBTi(aa_nat1139649368474441743_VEBTi(aa_opt762848812907537627_VEBTi(aa_Pro4022776811541674023_VEBTi(aa_VEB3655672368276914755_VEBTi(aa_arr737849036559383332_VEBTi(aa_nat7609637249401053881_VEBTi(aa_opt3216071514793242883_VEBTi(aTP_Lamm_aal(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_961
tff(fact_11137_ATP_Olambda__962,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat,Uug: option4927543243414619207at_nat] : aa_opt8585651768545592073_VEBTi(aa_Pro4062752345418257877_VEBTi(aa_VEB8867863569893255025_VEBTi(aa_arr3828982135699130450_VEBTi(aa_nat5629247741253790183_VEBTi(aa_opt9205748023383517873_VEBTi(aa_nat945706057799272946_VEBTi(aTP_Lamm_zo(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc7864503674192730076_VEBTi(aa_opt7507675525305149691_VEBTi(aa_Pro1234189438217840325_VEBTi(aa_VEB660175962507042495_VEBTi(aa_arr8367260621226259636_VEBTi(aa_nat5784496157376858101_VEBTi(aa_opt81624670976341313_VEBTi(aa_nat5602383413110735830_VEBTi(aTP_Lamm_zn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_962
tff(fact_11138_ATP_Olambda__963,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat,Uug: option4927543243414619207at_nat] : aa_opt8585651768545592073_VEBTi(aa_Pro4062752345418257877_VEBTi(aa_VEB8867863569893255025_VEBTi(aa_arr3828982135699130450_VEBTi(aa_nat5629247741253790183_VEBTi(aa_opt9205748023383517873_VEBTi(aa_nat945706057799272946_VEBTi(aTP_Lamm_wx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc7864503674192730076_VEBTi(aa_opt7507675525305149691_VEBTi(aa_Pro1234189438217840325_VEBTi(aa_VEB660175962507042495_VEBTi(aa_arr8367260621226259636_VEBTi(aa_nat5784496157376858101_VEBTi(aa_opt81624670976341313_VEBTi(aa_nat5602383413110735830_VEBTi(aTP_Lamm_ww(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_963
tff(fact_11139_ATP_Olambda__964,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat] : aa_opt1657869255803490932on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aa_VEB8598360222073186748on_nat(aTP_Lamm_pv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2606485630176857543on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aa_VEB3185010184023139490on_nat(aTP_Lamm_pu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_964
tff(fact_11140_ATP_Olambda__965,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat] : aa_opt1657869255803490932on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aa_VEB8598360222073186748on_nat(aTP_Lamm_pe(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2606485630176857543on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aa_VEB3185010184023139490on_nat(aTP_Lamm_pd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_965
tff(fact_11141_ATP_Olambda__966,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat] : aa_opt1657869255803490932on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aa_VEB8598360222073186748on_nat(aTP_Lamm_gv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2606485630176857543on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aa_VEB3185010184023139490on_nat(aTP_Lamm_gu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_966
tff(fact_11142_ATP_Olambda__967,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat] : aa_opt1657869255803490932on_nat(aa_VEB5714833192853586542on_nat(aa_arr7759188791488537571on_nat(aa_nat7273124837800733860on_nat(aa_opt1247461068388908528on_nat(aa_nat309676689105111045on_nat(aa_VEB8598360222073186748on_nat(aTP_Lamm_gg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc2606485630176857543on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aa_VEB3185010184023139490on_nat(aTP_Lamm_gf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_967
tff(fact_11143_ATP_Olambda__968,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: option4927543243414619207at_nat,Uug: nat] : aa_nat2075221075791484188on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aTP_Lamm_er(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc6124225815318652422on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aTP_Lamm_eq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_968
tff(fact_11144_ATP_Olambda__969,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: option4927543243414619207at_nat,Uug: nat] : aa_nat2075221075791484188on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aTP_Lamm_by(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = produc6124225815318652422on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aTP_Lamm_bx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_969
tff(fact_11145_ATP_Olambda__970,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: option_nat] :
      aa_opt9163293676379396302on_nat(aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_Pro2771509044941117526on_nat(aa_VEB8816156001103073488on_nat(aa_arr6004237943432320069on_nat(aa_nat4340965201858767110on_nat(aTP_Lamm_be(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $ite(
        ( ( Uuh != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Uue)),Uuh) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_predi,Uug),Uue),aa_nat9127621225095157975on_nat(aTP_Lamm_a(Uua),Uuf)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_predi,Uuc),Uuf),aa_Pro4885747571561342816on_nat(aa_arr7927785801258559901on_nat(aa_nat6148929329942773342on_nat(aTP_Lamm_bd(Uu),Uua),Uub),Uud)) ) ).

% ATP.lambda_970
tff(fact_11146_ATP_Olambda__971,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: product_unit] :
      aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_nat5674868329695104454Heap_o(aa_lis4631090546836902751Heap_o(aa_arr3453073379018417564Heap_o(aa_nat4764294232906884829Heap_o(aa_nat334188166746217062Heap_o(aTP_Lamm_tb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),Uug),
        heap_T7577940988442287570unit_o(refine_Imp_assert(( ( Uue = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),size_s6755466524823107622T_VEBT(Uud)) )),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aTP_Lamm_ta(Uu),Uuc),Uud),Uue),Uuf)),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) ) ).

% ATP.lambda_971
tff(fact_11147_ATP_Olambda__972,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: product_unit] :
      aa_Pro3379321066291785715Heap_o(aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_nat5674868329695104454Heap_o(aa_lis4631090546836902751Heap_o(aa_arr3453073379018417564Heap_o(aa_nat4764294232906884829Heap_o(aa_nat334188166746217062Heap_o(aTP_Lamm_oa(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),Uug),
        heap_T7577940988442287570unit_o(refine_Imp_assert(( ( Uue = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),size_s6755466524823107622T_VEBT(Uud)) )),aa_nat3811243159488243764Heap_o(aa_nat6178576981048102717Heap_o(aa_lis4519787659412806998Heap_o(aa_arr7619904339668570131Heap_o(aTP_Lamm_nz(Uu),Uuc),Uud),Uue),Uuf)),
        aa_o_H7399033255965945470Heap_o(heap_Time_return_o,$false) ) ).

% ATP.lambda_972
tff(fact_11148_ATP_Olambda__973,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aa_nat7200185421170774908on_nat(aTP_Lamm_ps(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aa_nat1315775135507255344on_nat(aTP_Lamm_pr(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uub) ).

% ATP.lambda_973
tff(fact_11149_ATP_Olambda__974,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aa_nat7200185421170774908on_nat(aTP_Lamm_pb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aa_nat1315775135507255344on_nat(aTP_Lamm_pa(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uub) ).

% ATP.lambda_974
tff(fact_11150_ATP_Olambda__975,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aa_nat7200185421170774908on_nat(aTP_Lamm_gs(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aa_nat1315775135507255344on_nat(aTP_Lamm_gr(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uub) ).

% ATP.lambda_975
tff(fact_11151_ATP_Olambda__976,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aa_nat7200185421170774908on_nat(aTP_Lamm_gd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = case_o8344607093967974880at_nat(heap_T3487192422709364219on_nat(none_nat),aa_VEB6310895951922269862on_nat(aa_lis7455679124549436279on_nat(aa_VEB7601412409763292529on_nat(aa_arr3206634112104793958on_nat(aa_nat2842375153920543911on_nat(aa_nat1315775135507255344on_nat(aTP_Lamm_gc(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uub) ).

% ATP.lambda_976
tff(fact_11152_ATP_Olambda__977,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: product_prod_nat_nat,Uuh: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_Pro8034723660955798012_VEBTi(aa_VEB4262372667711301016_VEBTi(aa_arr9036764603281787897_VEBTi(aa_nat2771851207891374350_VEBTi(aa_opt7594342031245826264_VEBTi(aa_nat4475881176255417753_VEBTi(aa_VEB8590999954632166712_VEBTi(aTP_Lamm_zp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = produc2624314226134418078_VEBTi(aa_Pro4062752345418257877_VEBTi(aa_VEB8867863569893255025_VEBTi(aa_arr3828982135699130450_VEBTi(aa_nat5629247741253790183_VEBTi(aa_opt9205748023383517873_VEBTi(aa_nat945706057799272946_VEBTi(aTP_Lamm_zo(Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),vEBT_c634343235235684882T_VEBT(aTP_Lamm_az,aTP_Lamm_ba,Uua)) ).

% ATP.lambda_977
tff(fact_11153_ATP_Olambda__978,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: product_prod_nat_nat,Uuh: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_Pro8034723660955798012_VEBTi(aa_VEB4262372667711301016_VEBTi(aa_arr9036764603281787897_VEBTi(aa_nat2771851207891374350_VEBTi(aa_opt7594342031245826264_VEBTi(aa_nat4475881176255417753_VEBTi(aa_VEB8590999954632166712_VEBTi(aTP_Lamm_wy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = produc2624314226134418078_VEBTi(aa_Pro4062752345418257877_VEBTi(aa_VEB8867863569893255025_VEBTi(aa_arr3828982135699130450_VEBTi(aa_nat5629247741253790183_VEBTi(aa_opt9205748023383517873_VEBTi(aa_nat945706057799272946_VEBTi(aTP_Lamm_wx(Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),vEBT_c634343235235684882T_VEBT(aTP_Lamm_az,aTP_Lamm_ba,Uua)) ).

% ATP.lambda_978
tff(fact_11154_ATP_Olambda__979,axiom,
    ! [Uu: nat,Uua: vEBT_VEBTi,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: $o,Uuh: array_VEBT_VEBTi] :
      aa_arr4839786837114051083_VEBTi(aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB6485260183185377513_VEBTi(aa_VEB953658682601944675_VEBTi(aTP_Lamm_zu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),(Uug)),Uuh) = heap_T1006145433769338483_VEBTi($ite((Uug),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(vEBT_V3964819847710782039nserti,Uub),Uua),Uuf),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Uua)),aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_nat3780430034849525329_VEBTi(aa_nat6565337644025228958_VEBTi(aTP_Lamm_wd(Uu),Uuc),Uud),Uue),Uuh)) ).

% ATP.lambda_979
tff(fact_11155_ATP_Olambda__980,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: $o,Uuh: array_VEBT_VEBTi] :
      aa_arr4839786837114051083_VEBTi(aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB8654527159822295598_VEBTi(aa_nat7794021720236223159_VEBTi(aTP_Lamm_xd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),(Uug)),Uuh) = heap_T1006145433769338483_VEBTi($ite((Uug),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(produc5159149307777246319_VEBTi(Uu),Uub),Uuf),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Uub)),aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_nat3780430034849525329_VEBTi(aa_nat6565337644025228958_VEBTi(aTP_Lamm_wd(Uua),Uuc),Uud),Uue),Uuh)) ).

% ATP.lambda_980
tff(fact_11156_ATP_Olambda__981,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: $o,Uuh: array_VEBT_VEBTi] :
      aa_arr4839786837114051083_VEBTi(aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB8654527159822295598_VEBTi(aa_nat7794021720236223159_VEBTi(aTP_Lamm_xs(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),(Uug)),Uuh) = heap_T1006145433769338483_VEBTi($ite((Uug),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(Uu,Uub),Uuf),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Uub)),aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_nat3780430034849525329_VEBTi(aa_nat6565337644025228958_VEBTi(aTP_Lamm_wd(Uua),Uuc),Uud),Uue),Uuh)) ).

% ATP.lambda_981
tff(fact_11157_ATP_Olambda__982,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: $o,Uuh: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB5075307909151313271_VEBTi(aa_arr5746817224660070252_VEBTi(aTP_Lamm_yg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),(Uug)),Uuh) = heap_T5099337393651448672_VEBTi(array_upd_VEBT_VEBTi(Uuf,Uuh,Uua),aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB8654527159822295598_VEBTi(aTP_Lamm_yf(Uu),Uub),Uuc),Uud),Uue),Uuf),(Uug))) ).

% ATP.lambda_982
tff(fact_11158_ATP_Olambda__983,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat] : aa_nat947733127081326238on_nat(aa_Pro4465521767867336112on_nat(aa_VEB6043830584257997615on_nat(aa_lis5985300925819340672on_nat(aa_VEB1019093884839074554on_nat(aa_arr1617095457630768239on_nat(aa_nat5782952277816596784on_nat(aa_nat7314605301869835321on_nat(aTP_Lamm_pq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB8035251902059208504on_nat(aa_lis5065134719111324425on_nat(aa_VEB1776269892753909763on_nat(aa_arr3452137061177157880on_nat(aa_nat8770382747262106937on_nat(aa_nat388734249307221954on_nat(aTP_Lamm_pp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_983
tff(fact_11159_ATP_Olambda__984,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat] : aa_nat947733127081326238on_nat(aa_Pro4465521767867336112on_nat(aa_VEB6043830584257997615on_nat(aa_lis5985300925819340672on_nat(aa_VEB1019093884839074554on_nat(aa_arr1617095457630768239on_nat(aa_nat5782952277816596784on_nat(aa_nat7314605301869835321on_nat(aTP_Lamm_gb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T8222160169144143993on_nat(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB8035251902059208504on_nat(aa_lis5065134719111324425on_nat(aa_VEB1776269892753909763on_nat(aa_arr3452137061177157880on_nat(aa_nat8770382747262106937on_nat(aa_nat388734249307221954on_nat(aTP_Lamm_ga(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_984
tff(fact_11160_ATP_Olambda__985,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB8966970847287376059_VEBTi(aa_arr4873757251330958748_VEBTi(aa_nat4224949603525894449_VEBTi(aTP_Lamm_yl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uuf),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB7864109228220129220_VEBTi(aa_arr7707872583627363365_VEBTi(aa_nat3757413148909559610_VEBTi(aTP_Lamm_yk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_985
tff(fact_11161_ATP_Olambda__986,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB8340470449056147762_VEBTi(aa_arr7322700162901495443_VEBTi(aa_nat4321419670308649640_VEBTi(aa_nat695344571205917557_VEBTi(aTP_Lamm_xz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uug),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB8966970847287376059_VEBTi(aa_arr4873757251330958748_VEBTi(aa_nat4224949603525894449_VEBTi(aa_nat7906690825329656190_VEBTi(aTP_Lamm_xy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_986
tff(fact_11162_ATP_Olambda__987,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB8340470449056147762_VEBTi(aa_arr7322700162901495443_VEBTi(aa_nat4321419670308649640_VEBTi(aa_nat695344571205917557_VEBTi(aTP_Lamm_xk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uug),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB8966970847287376059_VEBTi(aa_arr4873757251330958748_VEBTi(aa_nat4224949603525894449_VEBTi(aa_nat7906690825329656190_VEBTi(aTP_Lamm_xj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_987
tff(fact_11163_ATP_Olambda__988,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aTP_Lamm_ap(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),Uug),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB3438037858822377372on_nat(aa_lis3250066801158684525on_nat(aa_VEB8217410703897791847on_nat(aa_arr22699188990094940on_nat(aTP_Lamm_ao(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_988
tff(fact_11164_ATP_Olambda__989,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_ov(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uug),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB6116742807553843114on_nat(aa_lis2576702180724645807on_nat(aa_VEB3484668625599252427on_nat(aa_arr4903404720305071532on_nat(aa_nat907479165128101953on_nat(aTP_Lamm_ou(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_989
tff(fact_11165_ATP_Olambda__990,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_gm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uug),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB6116742807553843114on_nat(aa_lis2576702180724645807on_nat(aa_VEB3484668625599252427on_nat(aa_arr4903404720305071532on_nat(aa_nat907479165128101953on_nat(aTP_Lamm_gl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_990
tff(fact_11166_ATP_Olambda__991,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] :
      aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB3438037858822377372on_nat(aa_lis3250066801158684525on_nat(aa_VEB8217410703897791847on_nat(aa_arr22699188990094940on_nat(aTP_Lamm_ao(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= nth_VEBT_VEBT(Uuc,Uug),
        heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uuh),aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_Pro7328798252849467525on_nat(aa_VEB6402893080723908316on_nat(aa_VEB4808996286462526806on_nat(aa_arr4232376455131879627on_nat(aTP_Lamm_an(Uu),Uua),Uub),Uud),Uue),Uuf),Uug),Uuh),aktnode)) ) ).

% ATP.lambda_991
tff(fact_11167_ATP_Olambda__992,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] :
      aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB6116742807553843114on_nat(aa_lis2576702180724645807on_nat(aa_VEB3484668625599252427on_nat(aa_arr4903404720305071532on_nat(aa_nat907479165128101953on_nat(aTP_Lamm_ou(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= nth_VEBT_VEBT(Uud,Uug),
        heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Uuh),aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_VEB3020412023619788856on_nat(aa_VEB2867584020876442580on_nat(aa_arr7250883597209086773on_nat(aa_nat6287673228857293642on_nat(aTP_Lamm_ot(Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),aktnode)) ) ).

% ATP.lambda_992
tff(fact_11168_ATP_Olambda__993,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] :
      aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_VEB6116742807553843114on_nat(aa_lis2576702180724645807on_nat(aa_VEB3484668625599252427on_nat(aa_arr4903404720305071532on_nat(aa_nat907479165128101953on_nat(aTP_Lamm_gl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= nth_VEBT_VEBT(Uud,Uug),
        heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_maxti,Uuh),aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_VEB3020412023619788856on_nat(aa_VEB2867584020876442580on_nat(aa_arr7250883597209086773on_nat(aa_nat6287673228857293642on_nat(aTP_Lamm_gk(Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),aktnode)) ) ).

% ATP.lambda_993
tff(fact_11169_ATP_Olambda__994,axiom,
    ! [Uu: fun_Pr940977554174145406Heap_o,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: option4927543243414619207at_nat,Uuf: nat,Uug: list_VEBT_VEBT,Uuh: vEBT_VEBT] :
      aa_VEB8530157447395038106Heap_o(aa_lis4690688361886208363Heap_o(aa_nat3831830007592743808Heap_o(aa_opt3748239118003421002Heap_o(aa_arr581739785191926499Heap_o(aa_nat105892859602166008Heap_o(aa_opt8417070463701818562Heap_o(aa_nat8831994779944892035Heap_o(aTP_Lamm_th(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T7577940988442287570unit_o(refine_Imp_assert(( ( Uub = Uue )
          & ( Uuc = Uuf ) )),aa_lis4443831354813659076Heap_o(aa_arr8974760139355794305Heap_o(aa_nat808430574047188546Heap_o(aa_nat9059937328556229451Heap_o(aTP_Lamm_tg(Uu),Uua),Uuc),Uud),Uug)) ).

% ATP.lambda_994
tff(fact_11170_ATP_Olambda__995,axiom,
    ! [Uu: fun_VE8181444127144381288Heap_o,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: option4927543243414619207at_nat,Uuf: nat,Uug: list_VEBT_VEBT,Uuh: vEBT_VEBT] :
      aa_VEB8530157447395038106Heap_o(aa_lis4690688361886208363Heap_o(aa_nat3831830007592743808Heap_o(aa_opt3748239118003421002Heap_o(aa_arr581739785191926499Heap_o(aa_nat105892859602166008Heap_o(aa_opt8417070463701818562Heap_o(aa_nat8831994779944892035Heap_o(aTP_Lamm_og(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T7577940988442287570unit_o(refine_Imp_assert(( ( Uub = Uue )
          & ( Uuc = Uuf ) )),aa_lis4443831354813659076Heap_o(aa_arr8974760139355794305Heap_o(aa_nat808430574047188546Heap_o(aa_nat9059937328556229451Heap_o(aTP_Lamm_of(Uu),Uua),Uuc),Uud),Uug)) ).

% ATP.lambda_995
tff(fact_11171_ATP_Olambda__996,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB8035251902059208504on_nat(aa_lis5065134719111324425on_nat(aa_VEB1776269892753909763on_nat(aa_arr3452137061177157880on_nat(aa_nat8770382747262106937on_nat(aTP_Lamm_bt(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uug = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aTP_Lamm_bs(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_996
tff(fact_11172_ATP_Olambda__997,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_VEB4460395522175602758on_nat(aa_lis6278591930053894219on_nat(aa_VEB6094637369740064103on_nat(aa_arr5396080262017029960on_nat(aa_nat7337942549949891037on_nat(aa_nat8478446459054042410on_nat(aTP_Lamm_oy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uug = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aa_nat5704251136259731006on_nat(aTP_Lamm_ox(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_997
tff(fact_11173_ATP_Olambda__998,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_VEB4460395522175602758on_nat(aa_lis6278591930053894219on_nat(aa_VEB6094637369740064103on_nat(aa_arr5396080262017029960on_nat(aa_nat7337942549949891037on_nat(aa_nat8478446459054042410on_nat(aTP_Lamm_gp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uug = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aa_nat5704251136259731006on_nat(aTP_Lamm_go(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_998
tff(fact_11174_ATP_Olambda__999,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_el(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uug = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aTP_Lamm_ek(Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_999
tff(fact_11175_ATP_Olambda__1000,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aTP_Lamm_ar(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uug = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aTP_Lamm_aq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1000
tff(fact_11176_ATP_Olambda__1001,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi,Uug: vEBT_VEBT,Uuh: option_nat] : aa_opt9163293676379396302on_nat(aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_VEB3020412023619788856on_nat(aa_VEB2867584020876442580on_nat(aa_arr7250883597209086773on_nat(aTP_Lamm_eh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uuh = vEBT_vebt_maxt(Uug)),aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_VEB3297963669270601860on_nat(aa_VEB3046651646968523646on_nat(aa_arr2696250042600076787on_nat(aTP_Lamm_eg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1001
tff(fact_11177_ATP_Olambda__1002,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aTP_Lamm_aq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5999496708990702694on_nat(refine_Imp_assert(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uug),size_s6755466524823107622T_VEBT(Uuc))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aTP_Lamm_ap(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1002
tff(fact_11178_ATP_Olambda__1003,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_ow(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5999496708990702694on_nat(refine_Imp_assert(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uug),size_s6755466524823107622T_VEBT(Uud))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_ov(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1003
tff(fact_11179_ATP_Olambda__1004,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_gn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5999496708990702694on_nat(refine_Imp_assert(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uug),size_s6755466524823107622T_VEBT(Uud))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_gm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1004
tff(fact_11180_ATP_Olambda__1005,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB6809311486975643423_VEBTi(aa_arr4222437323582306304_VEBTi(aTP_Lamm_yi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T5998771940306268294_VEBTi(aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Uuh),aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB8848578851935055378_VEBTi(aa_arr7350257379194205831_VEBTi(aTP_Lamm_yh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1005
tff(fact_11181_ATP_Olambda__1006,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB6384125456408274455on_nat(aa_arr4162587047475849484on_nat(aa_nat5386638315231209293on_nat(aa_nat7318487011899192278on_nat(aTP_Lamm_sc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uuh),aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_Pro2771509044941117526on_nat(aa_VEB8816156001103073488on_nat(aa_arr6004237943432320069on_nat(aa_nat4340965201858767110on_nat(aa_nat1606558224950267919on_nat(aTP_Lamm_sb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1006
tff(fact_11182_ATP_Olambda__1007,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] : aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB6384125456408274455on_nat(aa_arr4162587047475849484on_nat(aa_nat5386638315231209293on_nat(aa_nat7318487011899192278on_nat(aTP_Lamm_gz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uuh),aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_Pro2771509044941117526on_nat(aa_VEB8816156001103073488on_nat(aa_arr6004237943432320069on_nat(aa_nat4340965201858767110on_nat(aa_nat1606558224950267919on_nat(aTP_Lamm_gy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1007
tff(fact_11183_ATP_Olambda__1008,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat,Uug: option4927543243414619207at_nat,Uuh: nat] : aa_nat7217600295869438385_VEBTi(aa_opt7507675525305149691_VEBTi(aa_Pro1234189438217840325_VEBTi(aa_VEB660175962507042495_VEBTi(aa_arr8367260621226259636_VEBTi(aa_nat5784496157376858101_VEBTi(aa_opt81624670976341313_VEBTi(aa_nat5602383413110735830_VEBTi(aTP_Lamm_zn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = produc9050507437146595227_VEBTi(aa_nat1139649368474441743_VEBTi(aa_opt762848812907537627_VEBTi(aa_Pro4022776811541674023_VEBTi(aa_VEB3655672368276914755_VEBTi(aa_arr737849036559383332_VEBTi(aa_nat7609637249401053881_VEBTi(aa_opt3216071514793242883_VEBTi(aa_nat8892849909372205124_VEBTi(aTP_Lamm_zm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1008
tff(fact_11184_ATP_Olambda__1009,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat,Uug: option4927543243414619207at_nat,Uuh: nat] : aa_nat7217600295869438385_VEBTi(aa_opt7507675525305149691_VEBTi(aa_Pro1234189438217840325_VEBTi(aa_VEB660175962507042495_VEBTi(aa_arr8367260621226259636_VEBTi(aa_nat5784496157376858101_VEBTi(aa_opt81624670976341313_VEBTi(aa_nat5602383413110735830_VEBTi(aTP_Lamm_ww(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = produc9050507437146595227_VEBTi(aa_nat1139649368474441743_VEBTi(aa_opt762848812907537627_VEBTi(aa_Pro4022776811541674023_VEBTi(aa_VEB3655672368276914755_VEBTi(aa_arr737849036559383332_VEBTi(aa_nat7609637249401053881_VEBTi(aa_opt3216071514793242883_VEBTi(aa_nat8892849909372205124_VEBTi(aTP_Lamm_wv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1009
tff(fact_11185_ATP_Olambda__1010,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat,Uuh: nat] : aa_nat2075221075791484188on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aa_VEB3185010184023139490on_nat(aTP_Lamm_pu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = produc6124225815318652422on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aa_VEB430102235636188558on_nat(aTP_Lamm_pt(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1010
tff(fact_11186_ATP_Olambda__1011,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat,Uuh: nat] : aa_nat2075221075791484188on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aa_VEB3185010184023139490on_nat(aTP_Lamm_pd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = produc6124225815318652422on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aa_VEB430102235636188558on_nat(aTP_Lamm_pc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1011
tff(fact_11187_ATP_Olambda__1012,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat,Uuh: nat] : aa_nat2075221075791484188on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aa_VEB3185010184023139490on_nat(aTP_Lamm_gu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = produc6124225815318652422on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aa_VEB430102235636188558on_nat(aTP_Lamm_gt(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1012
tff(fact_11188_ATP_Olambda__1013,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat,Uuh: nat] : aa_nat2075221075791484188on_nat(aa_opt8950567732158611302on_nat(aa_VEB8223317440372349570on_nat(aa_arr6272259649854615779on_nat(aa_nat261593688111926264on_nat(aa_opt4414834345254747970on_nat(aa_nat2224702164617786371on_nat(aa_VEB3185010184023139490on_nat(aTP_Lamm_gf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = produc6124225815318652422on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aa_VEB430102235636188558on_nat(aTP_Lamm_ge(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1013
tff(fact_11189_ATP_Olambda__1014,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi,Uug: vEBT_VEBT,Uuh: option_nat,Uui: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_VEB3297963669270601860on_nat(aa_VEB3046651646968523646on_nat(aa_arr2696250042600076787on_nat(aTP_Lamm_eg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = $ite(
        ( ( Uuh != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,Uud)),Uuh) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_succi,Uug),Uuf),Uud),aa_nat9127621225095157975on_nat(aTP_Lamm_a(Uu),Uue)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_succi,Uuc),Uub),Uue),aa_nat9127621225095157975on_nat(aa_VEB8264456441901112694on_nat(aa_arr7060518384107407923on_nat(aTP_Lamm_ef(Uu),Uua),Uuc),Uue)) ) ).

% ATP.lambda_1014
tff(fact_11190_ATP_Olambda__1015,axiom,
    ! [Uu: fun_Pr342922516890243977on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi,Uui: option_nat] :
      aa_opt9163293676379396302on_nat(aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_Pro2771509044941117526on_nat(aa_VEB8816156001103073488on_nat(aa_arr6004237943432320069on_nat(aa_nat4340965201858767110on_nat(aa_nat1606558224950267919on_nat(aTP_Lamm_sb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = $ite(
        ( ( Uui != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Uuf)),Uui) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(produc1489253303066280154on_nat(Uu),Uuh),Uuf),aa_nat9127621225095157975on_nat(aTP_Lamm_a(Uub),Uug)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(produc1489253303066280154on_nat(Uu),Uud),Uug),aa_Pro4885747571561342816on_nat(aa_arr7927785801258559901on_nat(aa_nat6148929329942773342on_nat(aTP_Lamm_bd(Uua),Uub),Uuc),Uue)) ) ).

% ATP.lambda_1015
tff(fact_11191_ATP_Olambda__1016,axiom,
    ! [Uu: fun_VE8516060933855021849on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi,Uui: option_nat] :
      aa_opt9163293676379396302on_nat(aa_VEB9161639482783362738on_nat(aa_nat6300058732544452155on_nat(aa_nat7962384762897155012on_nat(aa_Pro2771509044941117526on_nat(aa_VEB8816156001103073488on_nat(aa_arr6004237943432320069on_nat(aa_nat4340965201858767110on_nat(aa_nat1606558224950267919on_nat(aTP_Lamm_gy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = $ite(
        ( ( Uui != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Uuf)),Uui) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(Uu,Uuh),Uuf),aa_nat9127621225095157975on_nat(aTP_Lamm_a(Uub),Uug)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(Uu,Uud),Uug),aa_Pro4885747571561342816on_nat(aa_arr7927785801258559901on_nat(aa_nat6148929329942773342on_nat(aTP_Lamm_bd(Uua),Uub),Uuc),Uue)) ) ).

% ATP.lambda_1016
tff(fact_11192_ATP_Olambda__1017,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat,Uue: option4927543243414619207at_nat,Uuf: nat,Uug: list_VEBT_VEBT,Uuh: vEBT_VEBT,Uui: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_VEB3467812963353200047_VEBTi(aa_lis3316967545134462464_VEBTi(aa_nat7280733210739239061_VEBTi(aa_opt2515449710316153439_VEBTi(aa_Pro2414373876686838441_VEBTi(aa_VEB2192875243258015011_VEBTi(aa_arr5702273427728458776_VEBTi(aa_nat1667688020712374873_VEBTi(aTP_Lamm_aak(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = aa_Pro1395359565518501564_VEBTi(produc737604151543542771_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aTP_Lamm_aaj(Uu),Uua),Uub),Uuc),Uud),Uuf),Uug),Uuh)),the_Pr8591224930841456533at_nat(Uue)) ).

% ATP.lambda_1017
tff(fact_11193_ATP_Olambda__1018,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB681999334371813476_VEBTi(aa_lis3749794993760221801_VEBTi(aa_nat3264418630393705002_VEBTi(aa_VEB7195496841433988878_VEBTi(aa_arr953441552191549551_VEBTi(aa_nat1068536149936821380_VEBTi(aTP_Lamm_aah(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),Uuh),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uuh),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uu)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB3484995426995338861_VEBTi(aa_lis8107322492824698866_VEBTi(aa_nat5353086824354447155_VEBTi(aa_VEB4525472277563179415_VEBTi(aa_arr4621753813534489208_VEBTi(aa_nat369315079452798477_VEBTi(aTP_Lamm_aag(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1018
tff(fact_11194_ATP_Olambda__1019,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: $o,Uui: array_VEBT_VEBTi] :
      aa_arr4839786837114051083_VEBTi(aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB6485260183185377513_VEBTi(aa_VEB953658682601944675_VEBTi(aa_nat3348711059893815344_VEBTi(aTP_Lamm_yv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh)),Uui) = heap_T1006145433769338483_VEBTi($ite((Uuh),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(produc3311654011758026858_VEBTi(produc2164094337957399884_VEBTi(Uu)),Uuc),Uub),Uug),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Uub)),aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_nat3780430034849525329_VEBTi(aa_nat6565337644025228958_VEBTi(aTP_Lamm_wd(Uua),Uud),Uue),Uuf),Uui)) ).

% ATP.lambda_1019
tff(fact_11195_ATP_Olambda__1020,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: $o,Uui: array_VEBT_VEBTi] :
      aa_arr4839786837114051083_VEBTi(aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB6485260183185377513_VEBTi(aa_VEB953658682601944675_VEBTi(aa_nat3348711059893815344_VEBTi(aTP_Lamm_we(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh)),Uui) = heap_T1006145433769338483_VEBTi($ite((Uuh),aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(Uu,Uuc),Uub),Uug),aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,Uub)),aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_nat3780430034849525329_VEBTi(aa_nat6565337644025228958_VEBTi(aTP_Lamm_wd(Uua),Uud),Uue),Uuf),Uui)) ).

% ATP.lambda_1020
tff(fact_11196_ATP_Olambda__1021,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: $o,Uui: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB257407171870985852_VEBTi(aa_VEB2935853492131583640_VEBTi(aa_arr6469760262597239545_VEBTi(aTP_Lamm_zv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh)),Uui) = heap_T5099337393651448672_VEBTi(array_upd_VEBT_VEBTi(Uug,Uui,Uua),aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB6485260183185377513_VEBTi(aa_VEB953658682601944675_VEBTi(aTP_Lamm_zu(Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh))) ).

% ATP.lambda_1021
tff(fact_11197_ATP_Olambda__1022,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: $o,Uui: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB5075307909151313271_VEBTi(aa_arr5746817224660070252_VEBTi(aa_nat898395892396129965_VEBTi(aTP_Lamm_xt(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh)),Uui) = heap_T5099337393651448672_VEBTi(array_upd_VEBT_VEBTi(Uug,Uui,Uub),aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB8654527159822295598_VEBTi(aa_nat7794021720236223159_VEBTi(aTP_Lamm_xs(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),(Uuh))) ).

% ATP.lambda_1022
tff(fact_11198_ATP_Olambda__1023,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: $o,Uui: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB5075307909151313271_VEBTi(aa_arr5746817224660070252_VEBTi(aa_nat898395892396129965_VEBTi(aTP_Lamm_xe(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh)),Uui) = heap_T5099337393651448672_VEBTi(array_upd_VEBT_VEBTi(Uug,Uui,Uub),aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB8654527159822295598_VEBTi(aa_nat7794021720236223159_VEBTi(aTP_Lamm_xd(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),(Uuh))) ).

% ATP.lambda_1023
tff(fact_11199_ATP_Olambda__1024,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB8966970847287376059_VEBTi(aa_arr4873757251330958748_VEBTi(aa_nat4224949603525894449_VEBTi(aa_nat7906690825329656190_VEBTi(aTP_Lamm_xy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uug),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB7864109228220129220_VEBTi(aa_arr7707872583627363365_VEBTi(aa_nat3757413148909559610_VEBTi(aa_nat7878045268284587783_VEBTi(aTP_Lamm_xx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1024
tff(fact_11200_ATP_Olambda__1025,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB8966970847287376059_VEBTi(aa_arr4873757251330958748_VEBTi(aa_nat4224949603525894449_VEBTi(aa_nat7906690825329656190_VEBTi(aTP_Lamm_xj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uug),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB7864109228220129220_VEBTi(aa_arr7707872583627363365_VEBTi(aa_nat3757413148909559610_VEBTi(aa_nat7878045268284587783_VEBTi(aTP_Lamm_xi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1025
tff(fact_11201_ATP_Olambda__1026,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat,Uui: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aTP_Lamm_bq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uuh),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB3438037858822377372on_nat(aa_lis3250066801158684525on_nat(aa_VEB8217410703897791847on_nat(aa_arr22699188990094940on_nat(aa_nat3630391635445309085on_nat(aTP_Lamm_bp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1026
tff(fact_11202_ATP_Olambda__1027,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi,Uui: $o] : aa_o_H7577119287286566245_VEBTi(aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB8848578851935055378_VEBTi(aa_arr7350257379194205831_VEBTi(aTP_Lamm_yh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui)) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(vEBT_vebt_inserti,Uuh),Uuf),aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB5075307909151313271_VEBTi(aa_arr5746817224660070252_VEBTi(aTP_Lamm_yg(Uu),Uua),Uub),Uuc),Uud),Uue),Uug),(Uui))) ).

% ATP.lambda_1027
tff(fact_11203_ATP_Olambda__1028,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi] :
      aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB3438037858822377372on_nat(aa_lis3250066801158684525on_nat(aa_VEB8217410703897791847on_nat(aa_arr22699188990094940on_nat(aa_nat3630391635445309085on_nat(aTP_Lamm_bp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= nth_VEBT_VEBT(Uud,Uuh),
        heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uui),aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_Pro7328798252849467525on_nat(aa_VEB6402893080723908316on_nat(aa_VEB4808996286462526806on_nat(aa_arr4232376455131879627on_nat(aa_nat8394757134461757324on_nat(aTP_Lamm_bo(Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uui),aktnode)) ) ).

% ATP.lambda_1028
tff(fact_11204_ATP_Olambda__1029,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: option4927543243414619207at_nat,Uug: nat,Uuh: list_VEBT_VEBT,Uui: vEBT_VEBT] :
      aa_VEB3253283475358696116on_nat(aa_lis6893462050766468025on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aTP_Lamm_eq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(refine_Imp_assert(( ( Uuf = Uub )
          & ( Uug = Uuc )
          & aa_VEBT_VEBT_o(vEBT_is_Node,Uu) )),aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aTP_Lamm_ep(Uua),Uub),Uuc),Uud),Uue),Uuh),Uui)) ).

% ATP.lambda_1029
tff(fact_11205_ATP_Olambda__1030,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: option4927543243414619207at_nat,Uug: nat,Uuh: list_VEBT_VEBT,Uui: vEBT_VEBT] :
      aa_VEB3253283475358696116on_nat(aa_lis6893462050766468025on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aTP_Lamm_bx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(refine_Imp_assert(( ( Uuf = Uub )
          & ( Uug = Uuc )
          & aa_VEBT_VEBT_o(vEBT_is_Node,Uu) )),aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aTP_Lamm_bw(Uua),Uub),Uuc),Uud),Uue),Uuh),Uui)) ).

% ATP.lambda_1030
tff(fact_11206_ATP_Olambda__1031,axiom,
    ! [Uu: nat,Uua: option4927543243414619207at_nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: option4927543243414619207at_nat,Uug: nat,Uuh: list_VEBT_VEBT,Uui: vEBT_VEBT] :
      aa_VEB5809408762104066889_VEBTi(aa_lis894883570680706766_VEBTi(aa_nat1139649368474441743_VEBTi(aa_opt762848812907537627_VEBTi(aa_Pro4022776811541674023_VEBTi(aa_VEB3655672368276914755_VEBTi(aa_arr737849036559383332_VEBTi(aa_nat7609637249401053881_VEBTi(aa_opt3216071514793242883_VEBTi(aTP_Lamm_aal(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T7982501707604696571_VEBTi(refine_Imp_assert(( ( Uua = Uuf )
          & ( Uub = Uug ) )),aa_VEB3467812963353200047_VEBTi(aa_lis3316967545134462464_VEBTi(aa_nat7280733210739239061_VEBTi(aa_opt2515449710316153439_VEBTi(aa_Pro2414373876686838441_VEBTi(aa_VEB2192875243258015011_VEBTi(aa_arr5702273427728458776_VEBTi(aa_nat1667688020712374873_VEBTi(aTP_Lamm_aak(Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1031
tff(fact_11207_ATP_Olambda__1032,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB8035251902059208504on_nat(aa_lis5065134719111324425on_nat(aa_VEB1776269892753909763on_nat(aa_arr3452137061177157880on_nat(aa_nat8770382747262106937on_nat(aa_nat388734249307221954on_nat(aTP_Lamm_pp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uuh = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_po(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1032
tff(fact_11208_ATP_Olambda__1033,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat] : aa_nat947733127081326238on_nat(aa_nat6020981482381522343on_nat(aa_Pro8919772687891686969on_nat(aa_VEB8035251902059208504on_nat(aa_lis5065134719111324425on_nat(aa_VEB1776269892753909763on_nat(aa_arr3452137061177157880on_nat(aa_nat8770382747262106937on_nat(aa_nat388734249307221954on_nat(aTP_Lamm_ga(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uuh = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_fz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1033
tff(fact_11209_ATP_Olambda__1034,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat,Uui: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aTP_Lamm_bs(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uuh = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uu),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aTP_Lamm_br(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1034
tff(fact_11210_ATP_Olambda__1035,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aa_nat5704251136259731006on_nat(aTP_Lamm_ox(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uuh = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_ow(Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1035
tff(fact_11211_ATP_Olambda__1036,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aa_nat5704251136259731006on_nat(aTP_Lamm_go(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uuh = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_VEB6005108705944495412on_nat(aa_lis8565807915274807301on_nat(aa_VEB8546404647211134847on_nat(aa_arr8208594633606790516on_nat(aa_nat1839186240859117749on_nat(aTP_Lamm_gn(Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1036
tff(fact_11212_ATP_Olambda__1037,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: product_prod_nat_nat,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: vEBT_VEBT,Uui: option_nat] : aa_opt9163293676379396302on_nat(aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_Pro7328798252849467525on_nat(aa_VEB6402893080723908316on_nat(aa_VEB4808996286462526806on_nat(aa_arr4232376455131879627on_nat(aTP_Lamm_an(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uui = vEBT_vebt_mint(Uuh)),aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_Pro3918843774365325431on_nat(aa_VEB7240715040874449526on_nat(aa_VEB3377150917682228498on_nat(aa_arr2747111264493904243on_nat(aTP_Lamm_am(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1037
tff(fact_11213_ATP_Olambda__1038,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: vEBT_VEBT,Uui: option_nat] : aa_opt9163293676379396302on_nat(aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_VEB3020412023619788856on_nat(aa_VEB2867584020876442580on_nat(aa_arr7250883597209086773on_nat(aa_nat6287673228857293642on_nat(aTP_Lamm_ot(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uui = vEBT_vebt_maxt(Uuh)),aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_VEB3297963669270601860on_nat(aa_VEB3046651646968523646on_nat(aa_arr2696250042600076787on_nat(aa_nat8399877474291611060on_nat(aTP_Lamm_os(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1038
tff(fact_11214_ATP_Olambda__1039,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: vEBT_VEBT,Uui: option_nat] : aa_opt9163293676379396302on_nat(aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_VEB3020412023619788856on_nat(aa_VEB2867584020876442580on_nat(aa_arr7250883597209086773on_nat(aa_nat6287673228857293642on_nat(aTP_Lamm_gk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uui = vEBT_vebt_maxt(Uuh)),aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_VEB3297963669270601860on_nat(aa_VEB3046651646968523646on_nat(aa_arr2696250042600076787on_nat(aa_nat8399877474291611060on_nat(aTP_Lamm_gj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1039
tff(fact_11215_ATP_Olambda__1040,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat,Uui: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aTP_Lamm_br(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5999496708990702694on_nat(refine_Imp_assert(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuh),size_s6755466524823107622T_VEBT(Uud))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aTP_Lamm_bq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1040
tff(fact_11216_ATP_Olambda__1041,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aTP_Lamm_aai(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uud)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB681999334371813476_VEBTi(aa_lis3749794993760221801_VEBTi(aa_nat3264418630393705002_VEBTi(aa_VEB7195496841433988878_VEBTi(aa_arr953441552191549551_VEBTi(aa_nat1068536149936821380_VEBTi(aTP_Lamm_aah(Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1041
tff(fact_11217_ATP_Olambda__1042,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: product_prod_nat_nat,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aTP_Lamm_aaj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uud)),aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aTP_Lamm_aai(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1042
tff(fact_11218_ATP_Olambda__1043,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB6809311486975643423_VEBTi(aa_arr4222437323582306304_VEBTi(aa_nat3616447687739732885_VEBTi(aTP_Lamm_xv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5998771940306268294_VEBTi(aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Uui),aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB8848578851935055378_VEBTi(aa_arr7350257379194205831_VEBTi(aa_nat728959005377815624_VEBTi(aTP_Lamm_xu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1043
tff(fact_11219_ATP_Olambda__1044,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB6809311486975643423_VEBTi(aa_arr4222437323582306304_VEBTi(aa_nat3616447687739732885_VEBTi(aTP_Lamm_xg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T5998771940306268294_VEBTi(aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Uui),aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB8848578851935055378_VEBTi(aa_arr7350257379194205831_VEBTi(aa_nat728959005377815624_VEBTi(aTP_Lamm_xf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1044
tff(fact_11220_ATP_Olambda__1045,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB7864109228220129220_VEBTi(aa_arr7707872583627363365_VEBTi(aa_nat3757413148909559610_VEBTi(aTP_Lamm_yk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_T844888390831797134_VEBTi(array_len_VEBT_VEBTi(Uub),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB7525848083004366413_VEBTi(aa_arr912658909913700910_VEBTi(aa_nat5309850715826091203_VEBTi(aTP_Lamm_yj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1045
tff(fact_11221_ATP_Olambda__1046,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB7525848083004366413_VEBTi(aa_arr912658909913700910_VEBTi(aa_nat5309850715826091203_VEBTi(aTP_Lamm_yj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uui),Uuj)
        & ~ ( ( Uu = Uud )
            | ( Uu = Uue ) ) ),
        heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uui),aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB6809311486975643423_VEBTi(aa_arr4222437323582306304_VEBTi(aTP_Lamm_yi(Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uui)),
        aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uud),Uue)),Uua,Uub),Uuc)) ) ).

% ATP.lambda_1046
tff(fact_11222_ATP_Olambda__1047,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: vEBT_VEBT,Uui: option_nat,Uuj: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_VEB3297963669270601860on_nat(aa_VEB3046651646968523646on_nat(aa_arr2696250042600076787on_nat(aa_nat8399877474291611060on_nat(aTP_Lamm_os(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $ite(
        ( ( Uui != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,Uue)),Uui) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(produc9121605208400662357on_nat(produc1757988346207259447on_nat(Uu)),Uuh),Uug),Uue),aa_nat9127621225095157975on_nat(aTP_Lamm_a(Uua),Uuf)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(produc9121605208400662357on_nat(produc1757988346207259447on_nat(Uu)),Uud),Uuc),Uuf),aa_nat9127621225095157975on_nat(aa_VEB8264456441901112694on_nat(aa_arr7060518384107407923on_nat(aTP_Lamm_ef(Uua),Uub),Uud),Uuf)) ) ).

% ATP.lambda_1047
tff(fact_11223_ATP_Olambda__1048,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: vEBT_VEBT,Uui: option_nat,Uuj: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_VEB3297963669270601860on_nat(aa_VEB3046651646968523646on_nat(aa_arr2696250042600076787on_nat(aa_nat8399877474291611060on_nat(aTP_Lamm_gj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $ite(
        ( ( Uui != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,aa_nat_option_nat(some_nat,Uue)),Uui) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(Uu,Uuh),Uug),Uue),aa_nat9127621225095157975on_nat(aTP_Lamm_a(Uua),Uuf)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(Uu,Uud),Uuc),Uuf),aa_nat9127621225095157975on_nat(aa_VEB8264456441901112694on_nat(aa_arr7060518384107407923on_nat(aTP_Lamm_ef(Uua),Uub),Uud),Uuf)) ) ).

% ATP.lambda_1048
tff(fact_11224_ATP_Olambda__1049,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: product_prod_nat_nat,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: vEBT_VEBT,Uui: option_nat,Uuj: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_Pro3918843774365325431on_nat(aa_VEB7240715040874449526on_nat(aa_VEB3377150917682228498on_nat(aa_arr2747111264493904243on_nat(aTP_Lamm_am(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $ite(
        ( ( Uui != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Uue)),Uui) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(f,Uuh),Uug),Uue),aa_nat9127621225095157975on_nat(aTP_Lamm_a(Uu),Uuf)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(f,Uuc),Uub),Uuf),aa_nat9127621225095157975on_nat(aa_Pro5040713308973091561on_nat(aa_VEB8830394031746293224on_nat(aa_arr3423563577619635969on_nat(aTP_Lamm_al(Uu),Uua),Uuc),Uud),Uuf)) ) ).

% ATP.lambda_1049
tff(fact_11225_ATP_Olambda__1050,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: option4927543243414619207at_nat,Uug: nat,Uuh: list_VEBT_VEBT,Uui: vEBT_VEBT,Uuj: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_VEB3467812963353200047_VEBTi(aa_lis3316967545134462464_VEBTi(aa_nat7280733210739239061_VEBTi(aa_opt2515449710316153439_VEBTi(aa_Pro2414373876686838441_VEBTi(aa_VEB2192875243258015011_VEBTi(aa_arr5702273427728458776_VEBTi(aa_nat1667688020712374873_VEBTi(aa_nat345951160182191842_VEBTi(aTP_Lamm_zl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = aa_Pro1395359565518501564_VEBTi(produc737604151543542771_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aa_nat4303739465877327980_VEBTi(aTP_Lamm_zk(Uu),Uua),Uub),Uuc),Uud),Uue),Uug),Uuh),Uui)),the_Pr8591224930841456533at_nat(Uuf)) ).

% ATP.lambda_1050
tff(fact_11226_ATP_Olambda__1051,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: option4927543243414619207at_nat,Uug: nat,Uuh: list_VEBT_VEBT,Uui: vEBT_VEBT,Uuj: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_VEB3467812963353200047_VEBTi(aa_lis3316967545134462464_VEBTi(aa_nat7280733210739239061_VEBTi(aa_opt2515449710316153439_VEBTi(aa_Pro2414373876686838441_VEBTi(aa_VEB2192875243258015011_VEBTi(aa_arr5702273427728458776_VEBTi(aa_nat1667688020712374873_VEBTi(aa_nat345951160182191842_VEBTi(aTP_Lamm_wu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = aa_Pro1395359565518501564_VEBTi(produc737604151543542771_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aa_nat4303739465877327980_VEBTi(aTP_Lamm_wt(Uu),Uua),Uub),Uuc),Uud),Uue),Uug),Uuh),Uui)),the_Pr8591224930841456533at_nat(Uuf)) ).

% ATP.lambda_1051
tff(fact_11227_ATP_Olambda__1052,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB681999334371813476_VEBTi(aa_lis3749794993760221801_VEBTi(aa_nat3264418630393705002_VEBTi(aa_VEB7195496841433988878_VEBTi(aa_arr953441552191549551_VEBTi(aa_nat1068536149936821380_VEBTi(aa_nat1714504438291150161_VEBTi(aTP_Lamm_zi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uui),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uui),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB3484995426995338861_VEBTi(aa_lis8107322492824698866_VEBTi(aa_nat5353086824354447155_VEBTi(aa_VEB4525472277563179415_VEBTi(aa_arr4621753813534489208_VEBTi(aa_nat369315079452798477_VEBTi(aa_nat3166855462611258970_VEBTi(aTP_Lamm_zh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1052
tff(fact_11228_ATP_Olambda__1053,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB681999334371813476_VEBTi(aa_lis3749794993760221801_VEBTi(aa_nat3264418630393705002_VEBTi(aa_VEB7195496841433988878_VEBTi(aa_arr953441552191549551_VEBTi(aa_nat1068536149936821380_VEBTi(aa_nat1714504438291150161_VEBTi(aTP_Lamm_wr(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uui),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uui),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB3484995426995338861_VEBTi(aa_lis8107322492824698866_VEBTi(aa_nat5353086824354447155_VEBTi(aa_VEB4525472277563179415_VEBTi(aa_arr4621753813534489208_VEBTi(aa_nat369315079452798477_VEBTi(aa_nat3166855462611258970_VEBTi(aTP_Lamm_wq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1053
tff(fact_11229_ATP_Olambda__1054,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB3484995426995338861_VEBTi(aa_lis8107322492824698866_VEBTi(aa_nat5353086824354447155_VEBTi(aa_VEB4525472277563179415_VEBTi(aa_arr4621753813534489208_VEBTi(aa_nat369315079452798477_VEBTi(aTP_Lamm_aag(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $let(
        xn2: nat,
        xn2:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),Uug),Uug,Uu),
        heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uu),Uuh),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uu),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uuh)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB2417709626833497078_VEBTi(aa_lis5541900872984878331_VEBTi(aa_nat770714458426680252_VEBTi(aa_VEB4484153586529426336_VEBTi(aa_arr8317680606763408385_VEBTi(aa_nat4438223700708794134_VEBTi(aTP_Lamm_aaf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uuh),Uui),Uuj),xn2)) ) ).

% ATP.lambda_1054
tff(fact_11230_ATP_Olambda__1055,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: $o] : aa_o_H7577119287286566245_VEBTi(aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB8848578851935055378_VEBTi(aa_arr7350257379194205831_VEBTi(aa_nat728959005377815624_VEBTi(aTP_Lamm_xf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),(Uuj)) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(produc5159149307777246319_VEBTi(Uu),Uui),Uug),aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB5075307909151313271_VEBTi(aa_arr5746817224660070252_VEBTi(aa_nat898395892396129965_VEBTi(aTP_Lamm_xe(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uuh),(Uuj))) ).

% ATP.lambda_1055
tff(fact_11231_ATP_Olambda__1056,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: $o,Uuj: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB257407171870985852_VEBTi(aa_VEB2935853492131583640_VEBTi(aa_arr6469760262597239545_VEBTi(aa_nat7574034614035128846_VEBTi(aTP_Lamm_yw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui)),Uuj) = heap_T5099337393651448672_VEBTi(array_upd_VEBT_VEBTi(Uuh,Uuj,Uub),aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB6485260183185377513_VEBTi(aa_VEB953658682601944675_VEBTi(aa_nat3348711059893815344_VEBTi(aTP_Lamm_yv(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui))) ).

% ATP.lambda_1056
tff(fact_11232_ATP_Olambda__1057,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: $o,Uuj: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB257407171870985852_VEBTi(aa_VEB2935853492131583640_VEBTi(aa_arr6469760262597239545_VEBTi(aa_nat7574034614035128846_VEBTi(aTP_Lamm_wf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui)),Uuj) = heap_T5099337393651448672_VEBTi(array_upd_VEBT_VEBTi(Uuh,Uuj,Uub),aa_o_f5513965820244355694_VEBTi(aa_nat6231872576868200367_VEBTi(aa_nat3271849308718382136_VEBTi(aa_nat4180741733914107969_VEBTi(aa_nat6590260363808168906_VEBTi(aa_VEB6485260183185377513_VEBTi(aa_VEB953658682601944675_VEBTi(aa_nat3348711059893815344_VEBTi(aTP_Lamm_we(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui))) ).

% ATP.lambda_1057
tff(fact_11233_ATP_Olambda__1058,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: $o] : aa_o_H7577119287286566245_VEBTi(aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB8848578851935055378_VEBTi(aa_arr7350257379194205831_VEBTi(aa_nat728959005377815624_VEBTi(aTP_Lamm_xu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),(Uuj)) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(Uu,Uui),Uug),aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB5075307909151313271_VEBTi(aa_arr5746817224660070252_VEBTi(aa_nat898395892396129965_VEBTi(aTP_Lamm_xt(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uuh),(Uuj))) ).

% ATP.lambda_1058
tff(fact_11234_ATP_Olambda__1059,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aTP_Lamm_zz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uua),Uui),aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB779792351083059546_VEBTi(aa_lis578081645931084383_VEBTi(aa_VEB3237245999891841275_VEBTi(aa_arr5756643615615923676_VEBTi(aTP_Lamm_zy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1059
tff(fact_11235_ATP_Olambda__1060,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_pm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uuc),Uui),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB3438037858822377372on_nat(aa_lis3250066801158684525on_nat(aa_VEB8217410703897791847on_nat(aa_arr22699188990094940on_nat(aa_nat3630391635445309085on_nat(aa_nat7077015166316254502on_nat(aTP_Lamm_pl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1060
tff(fact_11236_ATP_Olambda__1061,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_fx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T2868974464944644318on_nat(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uuc),Uui),aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB3438037858822377372on_nat(aa_lis3250066801158684525on_nat(aa_VEB8217410703897791847on_nat(aa_arr22699188990094940on_nat(aa_nat3630391635445309085on_nat(aa_nat7077015166316254502on_nat(aTP_Lamm_fw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1061
tff(fact_11237_ATP_Olambda__1062,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat,Uuj: vEBT_VEBTi] :
      aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB3438037858822377372on_nat(aa_lis3250066801158684525on_nat(aa_VEB8217410703897791847on_nat(aa_arr22699188990094940on_nat(aa_nat3630391635445309085on_nat(aa_nat7077015166316254502on_nat(aTP_Lamm_pl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= nth_VEBT_VEBT(Uue,Uui),
        heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uuj),aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_Pro7328798252849467525on_nat(aa_VEB6402893080723908316on_nat(aa_VEB4808996286462526806on_nat(aa_arr4232376455131879627on_nat(aa_nat8394757134461757324on_nat(aa_nat5552918743037330581on_nat(aTP_Lamm_pk(Uu),Uua),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj),aktnode)) ) ).

% ATP.lambda_1062
tff(fact_11238_ATP_Olambda__1063,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat,Uuj: vEBT_VEBTi] :
      aa_VEB2711320435582112505on_nat(aa_nat4669826966810537986on_nat(aa_nat107244183536939787on_nat(aa_Pro1663420989643744413on_nat(aa_VEB3438037858822377372on_nat(aa_lis3250066801158684525on_nat(aa_VEB8217410703897791847on_nat(aa_arr22699188990094940on_nat(aa_nat3630391635445309085on_nat(aa_nat7077015166316254502on_nat(aTP_Lamm_fw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= nth_VEBT_VEBT(Uue,Uui),
        heap_T3669509953089699273on_nat(aa_VEB2711320435582112505on_nat(vEBT_vebt_minti,Uuj),aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_Pro7328798252849467525on_nat(aa_VEB6402893080723908316on_nat(aa_VEB4808996286462526806on_nat(aa_arr4232376455131879627on_nat(aa_nat8394757134461757324on_nat(aa_nat5552918743037330581on_nat(aTP_Lamm_fv(Uu),Uua),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj),aktnode)) ) ).

% ATP.lambda_1063
tff(fact_11239_ATP_Olambda__1064,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat,Uuh: nat,Uui: list_VEBT_VEBT,Uuj: vEBT_VEBT] :
      aa_VEB3253283475358696116on_nat(aa_lis6893462050766468025on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aa_VEB430102235636188558on_nat(aTP_Lamm_pt(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(refine_Imp_assert(( ( Uug = Uuc )
          & ( Uuh = Uud )
          & aa_VEBT_VEBT_o(vEBT_is_Node,Uua) )),aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aa_nat7200185421170774908on_nat(aTP_Lamm_ps(Uu),Uub),Uuc),Uud),Uue),Uuf),Uui),Uuj)) ).

% ATP.lambda_1064
tff(fact_11240_ATP_Olambda__1065,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat,Uuh: nat,Uui: list_VEBT_VEBT,Uuj: vEBT_VEBT] :
      aa_VEB3253283475358696116on_nat(aa_lis6893462050766468025on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aa_VEB430102235636188558on_nat(aTP_Lamm_pc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(refine_Imp_assert(( ( Uug = Uuc )
          & ( Uuh = Uud )
          & aa_VEBT_VEBT_o(vEBT_is_Node,Uua) )),aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aa_nat7200185421170774908on_nat(aTP_Lamm_pb(Uu),Uub),Uuc),Uud),Uue),Uuf),Uui),Uuj)) ).

% ATP.lambda_1065
tff(fact_11241_ATP_Olambda__1066,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat,Uuh: nat,Uui: list_VEBT_VEBT,Uuj: vEBT_VEBT] :
      aa_VEB3253283475358696116on_nat(aa_lis6893462050766468025on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aa_VEB430102235636188558on_nat(aTP_Lamm_gt(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(refine_Imp_assert(( ( Uug = Uuc )
          & ( Uuh = Uud )
          & aa_VEBT_VEBT_o(vEBT_is_Node,Uua) )),aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aa_nat7200185421170774908on_nat(aTP_Lamm_gs(Uu),Uub),Uuc),Uud),Uue),Uuf),Uui),Uuj)) ).

% ATP.lambda_1066
tff(fact_11242_ATP_Olambda__1067,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: vEBT_VEBT,Uub: nat,Uuc: option4927543243414619207at_nat,Uud: nat,Uue: array_VEBT_VEBTi,Uuf: vEBT_VEBTi,Uug: option4927543243414619207at_nat,Uuh: nat,Uui: list_VEBT_VEBT,Uuj: vEBT_VEBT] :
      aa_VEB3253283475358696116on_nat(aa_lis6893462050766468025on_nat(aa_nat8553908474689004666on_nat(aa_opt7284239819431652166on_nat(aa_VEB7959562887951478976on_nat(aa_arr4020752153017264181on_nat(aa_nat3060827894968391926on_nat(aa_opt346861859059968706on_nat(aa_nat3229958882251406039on_nat(aa_VEB430102235636188558on_nat(aTP_Lamm_ge(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(refine_Imp_assert(( ( Uug = Uuc )
          & ( Uuh = Uud )
          & aa_VEBT_VEBT_o(vEBT_is_Node,Uua) )),aa_VEB1832073013541904922on_nat(aa_lis6328568131235813867on_nat(aa_VEB3884484516340040293on_nat(aa_arr5442694692229379418on_nat(aa_nat4425110079127750043on_nat(aa_opt7218664994477926375on_nat(aa_nat7200185421170774908on_nat(aTP_Lamm_gd(Uu),Uub),Uuc),Uud),Uue),Uuf),Uui),Uuj)) ).

% ATP.lambda_1067
tff(fact_11243_ATP_Olambda__1068,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat,Uug: option4927543243414619207at_nat,Uuh: nat,Uui: list_VEBT_VEBT,Uuj: vEBT_VEBT] :
      aa_VEB5809408762104066889_VEBTi(aa_lis894883570680706766_VEBTi(aa_nat1139649368474441743_VEBTi(aa_opt762848812907537627_VEBTi(aa_Pro4022776811541674023_VEBTi(aa_VEB3655672368276914755_VEBTi(aa_arr737849036559383332_VEBTi(aa_nat7609637249401053881_VEBTi(aa_opt3216071514793242883_VEBTi(aa_nat8892849909372205124_VEBTi(aTP_Lamm_zm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T7982501707604696571_VEBTi(refine_Imp_assert(( ( Uub = Uug )
          & ( Uuc = Uuh ) )),aa_VEB3467812963353200047_VEBTi(aa_lis3316967545134462464_VEBTi(aa_nat7280733210739239061_VEBTi(aa_opt2515449710316153439_VEBTi(aa_Pro2414373876686838441_VEBTi(aa_VEB2192875243258015011_VEBTi(aa_arr5702273427728458776_VEBTi(aa_nat1667688020712374873_VEBTi(aa_nat345951160182191842_VEBTi(aTP_Lamm_zl(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1068
tff(fact_11244_ATP_Olambda__1069,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: option4927543243414619207at_nat,Uuc: nat,Uud: array_VEBT_VEBTi,Uue: vEBT_VEBTi,Uuf: product_prod_nat_nat,Uug: option4927543243414619207at_nat,Uuh: nat,Uui: list_VEBT_VEBT,Uuj: vEBT_VEBT] :
      aa_VEB5809408762104066889_VEBTi(aa_lis894883570680706766_VEBTi(aa_nat1139649368474441743_VEBTi(aa_opt762848812907537627_VEBTi(aa_Pro4022776811541674023_VEBTi(aa_VEB3655672368276914755_VEBTi(aa_arr737849036559383332_VEBTi(aa_nat7609637249401053881_VEBTi(aa_opt3216071514793242883_VEBTi(aa_nat8892849909372205124_VEBTi(aTP_Lamm_wv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T7982501707604696571_VEBTi(refine_Imp_assert(( ( Uub = Uug )
          & ( Uuc = Uuh ) )),aa_VEB3467812963353200047_VEBTi(aa_lis3316967545134462464_VEBTi(aa_nat7280733210739239061_VEBTi(aa_opt2515449710316153439_VEBTi(aa_Pro2414373876686838441_VEBTi(aa_VEB2192875243258015011_VEBTi(aa_arr5702273427728458776_VEBTi(aa_nat1667688020712374873_VEBTi(aa_nat345951160182191842_VEBTi(aTP_Lamm_wu(Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1069
tff(fact_11245_ATP_Olambda__1070,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_po(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uui = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_pn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1070
tff(fact_11246_ATP_Olambda__1071,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_fz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uui = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uua),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_fy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1071
tff(fact_11247_ATP_Olambda__1072,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi,Uui: vEBT_VEBT,Uuj: option_nat] : aa_opt9163293676379396302on_nat(aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_Pro7328798252849467525on_nat(aa_VEB6402893080723908316on_nat(aa_VEB4808996286462526806on_nat(aa_arr4232376455131879627on_nat(aa_nat8394757134461757324on_nat(aTP_Lamm_bo(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uuj = vEBT_vebt_mint(Uui)),aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_Pro3918843774365325431on_nat(aa_VEB7240715040874449526on_nat(aa_VEB3377150917682228498on_nat(aa_arr2747111264493904243on_nat(aa_nat610287491858673288on_nat(aTP_Lamm_bn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1072
tff(fact_11248_ATP_Olambda__1073,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aTP_Lamm_aaa(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T7982501707604696571_VEBTi(refine_Imp_assert(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uui),size_s6755466524823107622T_VEBT(Uuc))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aTP_Lamm_zz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1073
tff(fact_11249_ATP_Olambda__1074,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_pn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(refine_Imp_assert(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uui),size_s6755466524823107622T_VEBT(Uue))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_pm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1074
tff(fact_11250_ATP_Olambda__1075,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: product_prod_nat_nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa_Pro451875010097498139on_nat(aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_fy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5999496708990702694on_nat(refine_Imp_assert(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uui),size_s6755466524823107622T_VEBT(Uue))),aa_nat3999351741935936176on_nat(aa_nat3263736196148507005on_nat(aa_Pro6111869828830443265on_nat(aa_VEB6942907905770183000on_nat(aa_lis7408950245444934749on_nat(aa_VEB6661098386149969529on_nat(aa_arr9010034949507926874on_nat(aa_nat5148509663038818543on_nat(aa_nat8204191389585037116on_nat(aTP_Lamm_fx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1075
tff(fact_11251_ATP_Olambda__1076,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: list_VEBT_VEBT,Uuh: vEBT_VEBT,Uui: nat,Uuj: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aa_nat4303739465877327980_VEBTi(aTP_Lamm_zj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB681999334371813476_VEBTi(aa_lis3749794993760221801_VEBTi(aa_nat3264418630393705002_VEBTi(aa_VEB7195496841433988878_VEBTi(aa_arr953441552191549551_VEBTi(aa_nat1068536149936821380_VEBTi(aa_nat1714504438291150161_VEBTi(aTP_Lamm_zi(Uu),Uua),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1076
tff(fact_11252_ATP_Olambda__1077,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: list_VEBT_VEBT,Uuh: vEBT_VEBT,Uui: nat,Uuj: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aa_nat4303739465877327980_VEBTi(aTP_Lamm_ws(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_snd_nat_nat,Uue)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_VEB681999334371813476_VEBTi(aa_lis3749794993760221801_VEBTi(aa_nat3264418630393705002_VEBTi(aa_VEB7195496841433988878_VEBTi(aa_arr953441552191549551_VEBTi(aa_nat1068536149936821380_VEBTi(aa_nat1714504438291150161_VEBTi(aTP_Lamm_wr(Uu),Uua),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1077
tff(fact_11253_ATP_Olambda__1078,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: list_VEBT_VEBT,Uuh: vEBT_VEBT,Uui: nat,Uuj: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aa_nat4303739465877327980_VEBTi(aTP_Lamm_zk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uue)),aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aa_nat4303739465877327980_VEBTi(aTP_Lamm_zj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1078
tff(fact_11254_ATP_Olambda__1079,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: product_prod_nat_nat,Uuf: nat,Uug: list_VEBT_VEBT,Uuh: vEBT_VEBT,Uui: nat,Uuj: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aa_nat4303739465877327980_VEBTi(aTP_Lamm_wt(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(heap_Time_return_nat,aa_Pro4243173770044397973at_nat(product_fst_nat_nat,Uue)),aa_nat4282640288315822012_VEBTi(aa_VEB2281408947416558043_VEBTi(aa_lis148658726372936800_VEBTi(aa_nat3438343216000400545_VEBTi(aa_Pro9203685154717219507_VEBTi(aa_VEB4942273269570985133_VEBTi(aa_arr2286534195499807138_VEBTi(aa_nat2259444989923833827_VEBTi(aa_nat4303739465877327980_VEBTi(aTP_Lamm_ws(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1079
tff(fact_11255_ATP_Olambda__1080,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB779792351083059546_VEBTi(aa_lis578081645931084383_VEBTi(aa_VEB3237245999891841275_VEBTi(aa_arr5756643615615923676_VEBTi(aTP_Lamm_zy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T5998771940306268294_VEBTi(aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Uuj),aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB6059947344347517207_VEBTi(aa_lis3097942893158148200_VEBTi(aa_VEB2760733029397411170_VEBTi(aa_arr2574505544232410071_VEBTi(aTP_Lamm_zx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1080
tff(fact_11256_ATP_Olambda__1081,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB7864109228220129220_VEBTi(aa_arr7707872583627363365_VEBTi(aa_nat3757413148909559610_VEBTi(aa_nat7878045268284587783_VEBTi(aTP_Lamm_xx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T844888390831797134_VEBTi(array_len_VEBT_VEBTi(Uuc),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB7525848083004366413_VEBTi(aa_arr912658909913700910_VEBTi(aa_nat5309850715826091203_VEBTi(aa_nat1165803401171981328_VEBTi(aTP_Lamm_xw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1081
tff(fact_11257_ATP_Olambda__1082,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB7864109228220129220_VEBTi(aa_arr7707872583627363365_VEBTi(aa_nat3757413148909559610_VEBTi(aa_nat7878045268284587783_VEBTi(aTP_Lamm_xi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_T844888390831797134_VEBTi(array_len_VEBT_VEBTi(Uuc),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB7525848083004366413_VEBTi(aa_arr912658909913700910_VEBTi(aa_nat5309850715826091203_VEBTi(aa_nat1165803401171981328_VEBTi(aTP_Lamm_xh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1082
tff(fact_11258_ATP_Olambda__1083,axiom,
    ! [Uu: fun_VE8118064890490510894_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB7525848083004366413_VEBTi(aa_arr912658909913700910_VEBTi(aa_nat5309850715826091203_VEBTi(aa_nat1165803401171981328_VEBTi(aTP_Lamm_xw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuj),Uuk)
        & ~ ( ( Uua = Uue )
            | ( Uua = Uuf ) ) ),
        heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uuc),Uuj),aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB6809311486975643423_VEBTi(aa_arr4222437323582306304_VEBTi(aa_nat3616447687739732885_VEBTi(aTP_Lamm_xv(Uu),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj)),
        aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uue),Uuf)),Uub,Uuc),Uud)) ) ).

% ATP.lambda_1083
tff(fact_11259_ATP_Olambda__1084,axiom,
    ! [Uu: fun_Pr4171499318505651870_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB7525848083004366413_VEBTi(aa_arr912658909913700910_VEBTi(aa_nat5309850715826091203_VEBTi(aa_nat1165803401171981328_VEBTi(aTP_Lamm_xh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuj),Uuk)
        & ~ ( ( Uua = Uue )
            | ( Uua = Uuf ) ) ),
        heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uuc),Uuj),aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB6809311486975643423_VEBTi(aa_arr4222437323582306304_VEBTi(aa_nat3616447687739732885_VEBTi(aTP_Lamm_xg(Uu),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj)),
        aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uue),Uuf)),Uub,Uuc),Uud)) ) ).

% ATP.lambda_1084
tff(fact_11260_ATP_Olambda__1085,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: product_prod_nat_nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi,Uui: vEBT_VEBT,Uuj: option_nat,Uuk: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_Pro3918843774365325431on_nat(aa_VEB7240715040874449526on_nat(aa_VEB3377150917682228498on_nat(aa_arr2747111264493904243on_nat(aa_nat610287491858673288on_nat(aTP_Lamm_bn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = $ite(
        ( ( Uuj != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Uuf)),Uuj) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_predi,Uui),Uuh),Uuf),aa_nat9127621225095157975on_nat(aTP_Lamm_a(Uua),Uug)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(vEBT_VEBT_vebt_predi,Uud),Uuc),Uug),aa_nat9127621225095157975on_nat(aa_Pro5040713308973091561on_nat(aa_VEB8830394031746293224on_nat(aa_arr3423563577619635969on_nat(aa_nat2635311902590537750on_nat(aTP_Lamm_bm(Uu),Uua),Uub),Uud),Uue),Uug)) ) ).

% ATP.lambda_1085
tff(fact_11261_ATP_Olambda__1086,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB3484995426995338861_VEBTi(aa_lis8107322492824698866_VEBTi(aa_nat5353086824354447155_VEBTi(aa_VEB4525472277563179415_VEBTi(aa_arr4621753813534489208_VEBTi(aa_nat369315079452798477_VEBTi(aa_nat3166855462611258970_VEBTi(aTP_Lamm_zh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = $let(
        xn2: nat,
        xn2:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uuh),Uuh,Uua),
        heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uui),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uui)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB2417709626833497078_VEBTi(aa_lis5541900872984878331_VEBTi(aa_nat770714458426680252_VEBTi(aa_VEB4484153586529426336_VEBTi(aa_arr8317680606763408385_VEBTi(aa_nat4438223700708794134_VEBTi(aa_nat6528503438221098723_VEBTi(aTP_Lamm_zg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uui),Uuj),Uuk),xn2)) ) ).

% ATP.lambda_1086
tff(fact_11262_ATP_Olambda__1087,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_VEB3484995426995338861_VEBTi(aa_lis8107322492824698866_VEBTi(aa_nat5353086824354447155_VEBTi(aa_VEB4525472277563179415_VEBTi(aa_arr4621753813534489208_VEBTi(aa_nat369315079452798477_VEBTi(aa_nat3166855462611258970_VEBTi(aTP_Lamm_wq(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = $let(
        xn2: nat,
        xn2:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uuh),Uuh,Uua),
        heap_T844888390831797134_VEBTi($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uui),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uua),aa_nat5821310369566504398ap_nat(heap_Time_return_nat,Uui)),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB2417709626833497078_VEBTi(aa_lis5541900872984878331_VEBTi(aa_nat770714458426680252_VEBTi(aa_VEB4484153586529426336_VEBTi(aa_arr8317680606763408385_VEBTi(aa_nat4438223700708794134_VEBTi(aa_nat6528503438221098723_VEBTi(aTP_Lamm_wp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uui),Uuj),Uuk),xn2)) ) ).

% ATP.lambda_1087
tff(fact_11263_ATP_Olambda__1088,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB2417709626833497078_VEBTi(aa_lis5541900872984878331_VEBTi(aa_nat770714458426680252_VEBTi(aa_VEB4484153586529426336_VEBTi(aa_arr8317680606763408385_VEBTi(aa_nat4438223700708794134_VEBTi(aTP_Lamm_aaf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uui),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB5310930937212039423_VEBTi(aa_lis7835404322308837764_VEBTi(aa_nat8538784969185189829_VEBTi(aa_VEB5145764472262921513_VEBTi(aa_arr6568412587313337610_VEBTi(aa_nat1553062435445860255_VEBTi(aTP_Lamm_aae(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk)) ).

% ATP.lambda_1088
tff(fact_11264_ATP_Olambda__1089,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aa_nat461013784042436401_VEBTi(aTP_Lamm_za(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uuj),aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB779792351083059546_VEBTi(aa_lis578081645931084383_VEBTi(aa_VEB3237245999891841275_VEBTi(aa_arr5756643615615923676_VEBTi(aa_nat7386715292082006385_VEBTi(aTP_Lamm_yz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1089
tff(fact_11265_ATP_Olambda__1090,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aa_nat461013784042436401_VEBTi(aTP_Lamm_wj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(array_nth_VEBT_VEBTi(Uub),Uuj),aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB779792351083059546_VEBTi(aa_lis578081645931084383_VEBTi(aa_VEB3237245999891841275_VEBTi(aa_arr5756643615615923676_VEBTi(aa_nat7386715292082006385_VEBTi(aTP_Lamm_wi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1090
tff(fact_11266_ATP_Olambda__1091,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: vEBT_VEBTi,Uuk: $o] : aa_o_H7577119287286566245_VEBTi(aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB6059947344347517207_VEBTi(aa_lis3097942893158148200_VEBTi(aa_VEB2760733029397411170_VEBTi(aa_arr2574505544232410071_VEBTi(aTP_Lamm_zx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),(Uuk)) = heap_T7982501707604696571_VEBTi(refine_Imp_assert((Uuk) = aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(Uuc,Uui))),aa_o_f3680854670239116947_VEBTi(aa_VEB318638452350941999_VEBTi(aa_nat250827245977626168_VEBTi(aa_nat3169258303071516481_VEBTi(aa_nat904988376872050634_VEBTi(aa_nat6815201885744029651_VEBTi(aa_nat6697094627763402588_VEBTi(aa_VEB2406964849791727995_VEBTi(aa_lis5794688975234795520_VEBTi(aa_VEB4139799945145921820_VEBTi(aa_arr7513636907291371389_VEBTi(aTP_Lamm_zw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),(Uuk))) ).

% ATP.lambda_1091
tff(fact_11267_ATP_Olambda__1092,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: vEBT_VEBT,Uuk: option_nat] : aa_opt9163293676379396302on_nat(aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_Pro7328798252849467525on_nat(aa_VEB6402893080723908316on_nat(aa_VEB4808996286462526806on_nat(aa_arr4232376455131879627on_nat(aa_nat8394757134461757324on_nat(aa_nat5552918743037330581on_nat(aTP_Lamm_pk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uuk = vEBT_vebt_mint(Uuj)),aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_Pro3918843774365325431on_nat(aa_VEB7240715040874449526on_nat(aa_VEB3377150917682228498on_nat(aa_arr2747111264493904243on_nat(aa_nat610287491858673288on_nat(aa_nat9025683123140566101on_nat(aTP_Lamm_pj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk)) ).

% ATP.lambda_1092
tff(fact_11268_ATP_Olambda__1093,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: vEBT_VEBT,Uuk: option_nat] : aa_opt9163293676379396302on_nat(aa_VEB3755160987679246317on_nat(aa_VEB5212316371209427943on_nat(aa_nat7793430520228346548on_nat(aa_nat8410684691732023425on_nat(aa_Pro7328798252849467525on_nat(aa_VEB6402893080723908316on_nat(aa_VEB4808996286462526806on_nat(aa_arr4232376455131879627on_nat(aa_nat8394757134461757324on_nat(aa_nat5552918743037330581on_nat(aTP_Lamm_fv(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T5999496708990702694on_nat(refine_Imp_assert(Uuk = vEBT_vebt_mint(Uuj)),aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_Pro3918843774365325431on_nat(aa_VEB7240715040874449526on_nat(aa_VEB3377150917682228498on_nat(aa_arr2747111264493904243on_nat(aa_nat610287491858673288on_nat(aa_nat9025683123140566101on_nat(aTP_Lamm_fu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk)) ).

% ATP.lambda_1093
tff(fact_11269_ATP_Olambda__1094,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aa_nat461013784042436401_VEBTi(aTP_Lamm_zb(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T7982501707604696571_VEBTi(refine_Imp_assert(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuj),size_s6755466524823107622T_VEBT(Uud))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aa_nat461013784042436401_VEBTi(aTP_Lamm_za(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1094
tff(fact_11270_ATP_Olambda__1095,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aa_nat461013784042436401_VEBTi(aTP_Lamm_wk(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T7982501707604696571_VEBTi(refine_Imp_assert(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuj),size_s6755466524823107622T_VEBT(Uud))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aa_nat461013784042436401_VEBTi(aTP_Lamm_wj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1095
tff(fact_11271_ATP_Olambda__1096,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB779792351083059546_VEBTi(aa_lis578081645931084383_VEBTi(aa_VEB3237245999891841275_VEBTi(aa_arr5756643615615923676_VEBTi(aa_nat7386715292082006385_VEBTi(aTP_Lamm_yz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T5998771940306268294_VEBTi(aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Uuk),aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB6059947344347517207_VEBTi(aa_lis3097942893158148200_VEBTi(aa_VEB2760733029397411170_VEBTi(aa_arr2574505544232410071_VEBTi(aa_nat2429737420813335448_VEBTi(aTP_Lamm_yy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk)) ).

% ATP.lambda_1096
tff(fact_11272_ATP_Olambda__1097,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi] : aa_VEB836174949890266638_VEBTi(aa_nat5661399468481021847_VEBTi(aa_nat5003802506003144480_VEBTi(aa_nat6388535185314696745_VEBTi(aa_nat3402329839695434930_VEBTi(aa_nat7160451295336493755_VEBTi(aa_VEB779792351083059546_VEBTi(aa_lis578081645931084383_VEBTi(aa_VEB3237245999891841275_VEBTi(aa_arr5756643615615923676_VEBTi(aa_nat7386715292082006385_VEBTi(aTP_Lamm_wi(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_T5998771940306268294_VEBTi(aa_VEB8194921341634505877Heap_o(vEBT_VEBT_minNulli,Uuk),aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB6059947344347517207_VEBTi(aa_lis3097942893158148200_VEBTi(aa_VEB2760733029397411170_VEBTi(aa_arr2574505544232410071_VEBTi(aa_nat2429737420813335448_VEBTi(aTP_Lamm_wh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk)) ).

% ATP.lambda_1097
tff(fact_11273_ATP_Olambda__1098,axiom,
    ! [Uu: fun_Pr633315466666964198on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: vEBT_VEBT,Uuk: option_nat,Uul: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_Pro3918843774365325431on_nat(aa_VEB7240715040874449526on_nat(aa_VEB3377150917682228498on_nat(aa_arr2747111264493904243on_nat(aa_nat610287491858673288on_nat(aa_nat9025683123140566101on_nat(aTP_Lamm_pj(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul) = $ite(
        ( ( Uuk != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Uug)),Uuk) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(produc9121605208400662357on_nat(produc1757988346207259447on_nat(Uu)),Uuj),Uui),Uug),aa_nat9127621225095157975on_nat(aTP_Lamm_a(Uub),Uuh)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(produc9121605208400662357on_nat(produc1757988346207259447on_nat(Uu)),Uue),Uud),Uuh),aa_nat9127621225095157975on_nat(aa_Pro5040713308973091561on_nat(aa_VEB8830394031746293224on_nat(aa_arr3423563577619635969on_nat(aa_nat2635311902590537750on_nat(aTP_Lamm_bm(Uua),Uub),Uuc),Uue),Uuf),Uuh)) ) ).

% ATP.lambda_1098
tff(fact_11274_ATP_Olambda__1099,axiom,
    ! [Uu: fun_VE3775134097089128360on_nat,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: vEBT_VEBT,Uuf: product_prod_nat_nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: vEBT_VEBT,Uuk: option_nat,Uul: product_unit] :
      aa_Pro451875010097498139on_nat(aa_opt5089197864053947520on_nat(aa_VEB2490029800597546679on_nat(aa_VEB1055358423767010003on_nat(aa_nat4144897467549153500on_nat(aa_nat7557481500408122597on_nat(aa_Pro3918843774365325431on_nat(aa_VEB7240715040874449526on_nat(aa_VEB3377150917682228498on_nat(aa_arr2747111264493904243on_nat(aa_nat610287491858673288on_nat(aa_nat9025683123140566101on_nat(aTP_Lamm_fu(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul) = $ite(
        ( ( Uuk != none_nat )
        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,aa_nat_option_nat(some_nat,Uug)),Uuk) ),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(Uu,Uuj),Uui),Uug),aa_nat9127621225095157975on_nat(aTP_Lamm_a(Uub),Uuh)),
        heap_T3669509953089699273on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(Uu,Uue),Uud),Uuh),aa_nat9127621225095157975on_nat(aa_Pro5040713308973091561on_nat(aa_VEB8830394031746293224on_nat(aa_arr3423563577619635969on_nat(aa_nat2635311902590537750on_nat(aTP_Lamm_bm(Uua),Uub),Uuc),Uue),Uuf),Uuh)) ) ).

% ATP.lambda_1099
tff(fact_11275_ATP_Olambda__1100,axiom,
    ! [Uu: nat,Uua: array_VEBT_VEBTi,Uub: vEBT_VEBTi,Uuc: list_VEBT_VEBT,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: vEBT_VEBTi,Uuk: $o,Uul: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_o_f3680854670239116947_VEBTi(aa_VEB318638452350941999_VEBTi(aa_nat250827245977626168_VEBTi(aa_nat3169258303071516481_VEBTi(aa_nat904988376872050634_VEBTi(aa_nat6815201885744029651_VEBTi(aa_nat6697094627763402588_VEBTi(aa_VEB2406964849791727995_VEBTi(aa_lis5794688975234795520_VEBTi(aa_VEB4139799945145921820_VEBTi(aa_arr7513636907291371389_VEBTi(aTP_Lamm_zw(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),(Uuk)),Uul) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(vEBT_V3964819847710782039nserti,nth_VEBT_VEBT(Uuc,Uui)),Uuj),Uuh),aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB257407171870985852_VEBTi(aa_VEB2935853492131583640_VEBTi(aa_arr6469760262597239545_VEBTi(aTP_Lamm_zv(Uu),Uua),Uub),Uud),Uue),Uuf),Uug),Uui),(Uuk))) ).

% ATP.lambda_1100
tff(fact_11276_ATP_Olambda__1101,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB2417709626833497078_VEBTi(aa_lis5541900872984878331_VEBTi(aa_nat770714458426680252_VEBTi(aa_VEB4484153586529426336_VEBTi(aa_arr8317680606763408385_VEBTi(aa_nat4438223700708794134_VEBTi(aa_nat6528503438221098723_VEBTi(aTP_Lamm_zg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uuj),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB5310930937212039423_VEBTi(aa_lis7835404322308837764_VEBTi(aa_nat8538784969185189829_VEBTi(aa_VEB5145764472262921513_VEBTi(aa_arr6568412587313337610_VEBTi(aa_nat1553062435445860255_VEBTi(aa_nat1501880811460822764_VEBTi(aTP_Lamm_zf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul)) ).

% ATP.lambda_1101
tff(fact_11277_ATP_Olambda__1102,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_VEB2417709626833497078_VEBTi(aa_lis5541900872984878331_VEBTi(aa_nat770714458426680252_VEBTi(aa_VEB4484153586529426336_VEBTi(aa_arr8317680606763408385_VEBTi(aa_nat4438223700708794134_VEBTi(aa_nat6528503438221098723_VEBTi(aTP_Lamm_wp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_lowi,Uuj),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB5310930937212039423_VEBTi(aa_lis7835404322308837764_VEBTi(aa_nat8538784969185189829_VEBTi(aa_VEB5145764472262921513_VEBTi(aa_arr6568412587313337610_VEBTi(aa_nat1553062435445860255_VEBTi(aa_nat1501880811460822764_VEBTi(aTP_Lamm_wo(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul)) ).

% ATP.lambda_1102
tff(fact_11278_ATP_Olambda__1103,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB5310930937212039423_VEBTi(aa_lis7835404322308837764_VEBTi(aa_nat8538784969185189829_VEBTi(aa_VEB5145764472262921513_VEBTi(aa_arr6568412587313337610_VEBTi(aa_nat1553062435445860255_VEBTi(aTP_Lamm_aae(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul) = heap_T7982501707604696571_VEBTi(refine_Imp_assert(Uul = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uuj),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uud),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_nat5640937189053280966_VEBTi(aa_VEB5864305904528209917_VEBTi(aa_lis8083850882120436046_VEBTi(aa_nat6903794420489964771_VEBTi(aa_VEB7377128652718434837_VEBTi(aa_arr6679936031850374666_VEBTi(aa_nat7444525983491697995_VEBTi(aTP_Lamm_aad(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul)) ).

% ATP.lambda_1103
tff(fact_11279_ATP_Olambda__1104,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi,Uul: $o] : aa_o_H7577119287286566245_VEBTi(aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB6059947344347517207_VEBTi(aa_lis3097942893158148200_VEBTi(aa_VEB2760733029397411170_VEBTi(aa_arr2574505544232410071_VEBTi(aa_nat2429737420813335448_VEBTi(aTP_Lamm_yy(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul)) = heap_T7982501707604696571_VEBTi(refine_Imp_assert((Uul) = aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(Uud,Uuj))),aa_o_f3680854670239116947_VEBTi(aa_VEB318638452350941999_VEBTi(aa_nat250827245977626168_VEBTi(aa_nat3169258303071516481_VEBTi(aa_nat904988376872050634_VEBTi(aa_nat6815201885744029651_VEBTi(aa_nat6697094627763402588_VEBTi(aa_VEB2406964849791727995_VEBTi(aa_lis5794688975234795520_VEBTi(aa_VEB4139799945145921820_VEBTi(aa_arr7513636907291371389_VEBTi(aa_nat1261261390489106578_VEBTi(aTP_Lamm_yx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul))) ).

% ATP.lambda_1104
tff(fact_11280_ATP_Olambda__1105,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi,Uul: $o] : aa_o_H7577119287286566245_VEBTi(aa_VEB2625734152646192863_VEBTi(aa_nat3994193416293967532_VEBTi(aa_nat4708659386357353849_VEBTi(aa_nat6007137331526128070_VEBTi(aa_nat4759958082780230547_VEBTi(aa_nat5669347316383757024_VEBTi(aa_VEB6059947344347517207_VEBTi(aa_lis3097942893158148200_VEBTi(aa_VEB2760733029397411170_VEBTi(aa_arr2574505544232410071_VEBTi(aa_nat2429737420813335448_VEBTi(aTP_Lamm_wh(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul)) = heap_T7982501707604696571_VEBTi(refine_Imp_assert((Uul) = aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(Uud,Uuj))),aa_o_f3680854670239116947_VEBTi(aa_VEB318638452350941999_VEBTi(aa_nat250827245977626168_VEBTi(aa_nat3169258303071516481_VEBTi(aa_nat904988376872050634_VEBTi(aa_nat6815201885744029651_VEBTi(aa_nat6697094627763402588_VEBTi(aa_VEB2406964849791727995_VEBTi(aa_lis5794688975234795520_VEBTi(aa_VEB4139799945145921820_VEBTi(aa_arr7513636907291371389_VEBTi(aa_nat1261261390489106578_VEBTi(aTP_Lamm_wg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul))) ).

% ATP.lambda_1105
tff(fact_11281_ATP_Olambda__1106,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi,Uul: $o,Uum: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_o_f3680854670239116947_VEBTi(aa_VEB318638452350941999_VEBTi(aa_nat250827245977626168_VEBTi(aa_nat3169258303071516481_VEBTi(aa_nat904988376872050634_VEBTi(aa_nat6815201885744029651_VEBTi(aa_nat6697094627763402588_VEBTi(aa_VEB2406964849791727995_VEBTi(aa_lis5794688975234795520_VEBTi(aa_VEB4139799945145921820_VEBTi(aa_arr7513636907291371389_VEBTi(aa_nat1261261390489106578_VEBTi(aTP_Lamm_yx(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul)),Uum) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(produc3311654011758026858_VEBTi(produc2164094337957399884_VEBTi(Uu)),nth_VEBT_VEBT(Uud,Uuj)),Uuk),Uui),aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB257407171870985852_VEBTi(aa_VEB2935853492131583640_VEBTi(aa_arr6469760262597239545_VEBTi(aa_nat7574034614035128846_VEBTi(aTP_Lamm_yw(Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uuj),(Uul))) ).

% ATP.lambda_1106
tff(fact_11282_ATP_Olambda__1107,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: list_VEBT_VEBT,Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi,Uul: $o,Uum: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_o_f3680854670239116947_VEBTi(aa_VEB318638452350941999_VEBTi(aa_nat250827245977626168_VEBTi(aa_nat3169258303071516481_VEBTi(aa_nat904988376872050634_VEBTi(aa_nat6815201885744029651_VEBTi(aa_nat6697094627763402588_VEBTi(aa_VEB2406964849791727995_VEBTi(aa_lis5794688975234795520_VEBTi(aa_VEB4139799945145921820_VEBTi(aa_arr7513636907291371389_VEBTi(aa_nat1261261390489106578_VEBTi(aTP_Lamm_wg(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul)),Uum) = heap_T1006145433769338483_VEBTi(aa_nat5408703056993926195_VEBTi(aa_VEB4833742232818206615_VEBTi(aa_VEB6292075424854053302_VEBTi(Uu,nth_VEBT_VEBT(Uud,Uuj)),Uuk),Uui),aa_o_f1343430628605260489_VEBTi(aa_nat4147387395820660958_VEBTi(aa_nat2236254811235047083_VEBTi(aa_nat5995673692660408824_VEBTi(aa_nat4988899645790341829_VEBTi(aa_VEB257407171870985852_VEBTi(aa_VEB2935853492131583640_VEBTi(aa_arr6469760262597239545_VEBTi(aa_nat7574034614035128846_VEBTi(aTP_Lamm_wf(Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uuj),(Uul))) ).

% ATP.lambda_1107
tff(fact_11283_ATP_Olambda__1108,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_nat5640937189053280966_VEBTi(aa_VEB5864305904528209917_VEBTi(aa_lis8083850882120436046_VEBTi(aa_nat6903794420489964771_VEBTi(aa_VEB7377128652718434837_VEBTi(aa_arr6679936031850374666_VEBTi(aa_nat7444525983491697995_VEBTi(aTP_Lamm_aad(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uui),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB2726135844941024648_VEBTi(aa_lis3478650067825548685_VEBTi(aa_nat7061688277739565902_VEBTi(aa_VEB3820308522731820594_VEBTi(aa_arr165755620540720531_VEBTi(aa_nat5755912138998335400_VEBTi(aTP_Lamm_aac(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul)) ).

% ATP.lambda_1108
tff(fact_11284_ATP_Olambda__1109,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB5310930937212039423_VEBTi(aa_lis7835404322308837764_VEBTi(aa_nat8538784969185189829_VEBTi(aa_VEB5145764472262921513_VEBTi(aa_arr6568412587313337610_VEBTi(aa_nat1553062435445860255_VEBTi(aa_nat1501880811460822764_VEBTi(aTP_Lamm_zf(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum) = heap_T7982501707604696571_VEBTi(refine_Imp_assert(Uum = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uuk),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uue),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_nat5640937189053280966_VEBTi(aa_VEB5864305904528209917_VEBTi(aa_lis8083850882120436046_VEBTi(aa_nat6903794420489964771_VEBTi(aa_VEB7377128652718434837_VEBTi(aa_arr6679936031850374666_VEBTi(aa_nat7444525983491697995_VEBTi(aa_nat8034297438400619220_VEBTi(aTP_Lamm_ze(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum)) ).

% ATP.lambda_1109
tff(fact_11285_ATP_Olambda__1110,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_VEB5310930937212039423_VEBTi(aa_lis7835404322308837764_VEBTi(aa_nat8538784969185189829_VEBTi(aa_VEB5145764472262921513_VEBTi(aa_arr6568412587313337610_VEBTi(aa_nat1553062435445860255_VEBTi(aa_nat1501880811460822764_VEBTi(aTP_Lamm_wo(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum) = heap_T7982501707604696571_VEBTi(refine_Imp_assert(Uum = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uuk),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uue),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_nat5640937189053280966_VEBTi(aa_VEB5864305904528209917_VEBTi(aa_lis8083850882120436046_VEBTi(aa_nat6903794420489964771_VEBTi(aa_VEB7377128652718434837_VEBTi(aa_arr6679936031850374666_VEBTi(aa_nat7444525983491697995_VEBTi(aa_nat8034297438400619220_VEBTi(aTP_Lamm_wn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum)) ).

% ATP.lambda_1110
tff(fact_11286_ATP_Olambda__1111,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB2726135844941024648_VEBTi(aa_lis3478650067825548685_VEBTi(aa_nat7061688277739565902_VEBTi(aa_VEB3820308522731820594_VEBTi(aa_arr165755620540720531_VEBTi(aa_nat5755912138998335400_VEBTi(aTP_Lamm_aac(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum) = heap_T844888390831797134_VEBTi(array_len_VEBT_VEBTi(Uub),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_nat354071531552010866_VEBTi(aa_VEB545764574255792529_VEBTi(aa_lis7527251260247064854_VEBTi(aa_nat2109960713227968087_VEBTi(aa_VEB4528852985946928827_VEBTi(aa_arr8719208259821620636_VEBTi(aa_nat4030471164133998385_VEBTi(aTP_Lamm_aab(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum)) ).

% ATP.lambda_1111
tff(fact_11287_ATP_Olambda__1112,axiom,
    ! [Uu: nat,Uua: nat,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi,Uud: nat,Uue: list_VEBT_VEBT,Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_nat354071531552010866_VEBTi(aa_VEB545764574255792529_VEBTi(aa_lis7527251260247064854_VEBTi(aa_nat2109960713227968087_VEBTi(aa_VEB4528852985946928827_VEBTi(aa_arr8719208259821620636_VEBTi(aa_nat4030471164133998385_VEBTi(aTP_Lamm_aab(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uum),Uun)
        & ~ ( ( Uu = Uug )
            | ( Uu = Uuh ) ) ),
        heap_T7982501707604696571_VEBTi(refine_Imp_assert(Uum = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uuj),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uud),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aTP_Lamm_aaa(Uua),Uub),Uuc),Uue),Uuf),Uuh),Uui),Uuk),Uul),Uum)),
        aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uug),Uuh)),Uua,Uub),Uuc)) ) ).

% ATP.lambda_1112
tff(fact_11288_ATP_Olambda__1113,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_nat5640937189053280966_VEBTi(aa_VEB5864305904528209917_VEBTi(aa_lis8083850882120436046_VEBTi(aa_nat6903794420489964771_VEBTi(aa_VEB7377128652718434837_VEBTi(aa_arr6679936031850374666_VEBTi(aa_nat7444525983491697995_VEBTi(aa_nat8034297438400619220_VEBTi(aTP_Lamm_ze(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uuj),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB2726135844941024648_VEBTi(aa_lis3478650067825548685_VEBTi(aa_nat7061688277739565902_VEBTi(aa_VEB3820308522731820594_VEBTi(aa_arr165755620540720531_VEBTi(aa_nat5755912138998335400_VEBTi(aa_nat7935733879562905205_VEBTi(aTP_Lamm_zd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum)) ).

% ATP.lambda_1113
tff(fact_11289_ATP_Olambda__1114,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: product_unit] : aa_Pro3581650104493980720_VEBTi(aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_nat5640937189053280966_VEBTi(aa_VEB5864305904528209917_VEBTi(aa_lis8083850882120436046_VEBTi(aa_nat6903794420489964771_VEBTi(aa_VEB7377128652718434837_VEBTi(aa_arr6679936031850374666_VEBTi(aa_nat7444525983491697995_VEBTi(aa_nat8034297438400619220_VEBTi(aTP_Lamm_wn(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun) = heap_T844888390831797134_VEBTi(aa_nat5821310369566504398ap_nat(aa_nat110758267889697239ap_nat(vEBT_VEBT_highi,Uuj),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB2726135844941024648_VEBTi(aa_lis3478650067825548685_VEBTi(aa_nat7061688277739565902_VEBTi(aa_VEB3820308522731820594_VEBTi(aa_arr165755620540720531_VEBTi(aa_nat5755912138998335400_VEBTi(aa_nat7935733879562905205_VEBTi(aTP_Lamm_wm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum)) ).

% ATP.lambda_1114
tff(fact_11290_ATP_Olambda__1115,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB2726135844941024648_VEBTi(aa_lis3478650067825548685_VEBTi(aa_nat7061688277739565902_VEBTi(aa_VEB3820308522731820594_VEBTi(aa_arr165755620540720531_VEBTi(aa_nat5755912138998335400_VEBTi(aa_nat7935733879562905205_VEBTi(aTP_Lamm_zd(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun) = heap_T844888390831797134_VEBTi(array_len_VEBT_VEBTi(Uuc),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_nat354071531552010866_VEBTi(aa_VEB545764574255792529_VEBTi(aa_lis7527251260247064854_VEBTi(aa_nat2109960713227968087_VEBTi(aa_VEB4528852985946928827_VEBTi(aa_arr8719208259821620636_VEBTi(aa_nat4030471164133998385_VEBTi(aa_nat552572580330102142_VEBTi(aTP_Lamm_zc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun)) ).

% ATP.lambda_1115
tff(fact_11291_ATP_Olambda__1116,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: nat] : aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_VEB2726135844941024648_VEBTi(aa_lis3478650067825548685_VEBTi(aa_nat7061688277739565902_VEBTi(aa_VEB3820308522731820594_VEBTi(aa_arr165755620540720531_VEBTi(aa_nat5755912138998335400_VEBTi(aa_nat7935733879562905205_VEBTi(aTP_Lamm_wm(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun) = heap_T844888390831797134_VEBTi(array_len_VEBT_VEBTi(Uuc),aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_nat354071531552010866_VEBTi(aa_VEB545764574255792529_VEBTi(aa_lis7527251260247064854_VEBTi(aa_nat2109960713227968087_VEBTi(aa_VEB4528852985946928827_VEBTi(aa_arr8719208259821620636_VEBTi(aa_nat4030471164133998385_VEBTi(aa_nat552572580330102142_VEBTi(aTP_Lamm_wl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun)) ).

% ATP.lambda_1116
tff(fact_11292_ATP_Olambda__1117,axiom,
    ! [Uu: fun_Pr4404027214751104635_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: nat,Uuo: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_nat354071531552010866_VEBTi(aa_VEB545764574255792529_VEBTi(aa_lis7527251260247064854_VEBTi(aa_nat2109960713227968087_VEBTi(aa_VEB4528852985946928827_VEBTi(aa_arr8719208259821620636_VEBTi(aa_nat4030471164133998385_VEBTi(aa_nat552572580330102142_VEBTi(aTP_Lamm_zc(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun),Uuo) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uun),Uuo)
        & ~ ( ( Uua = Uuh )
            | ( Uua = Uui ) ) ),
        heap_T7982501707604696571_VEBTi(refine_Imp_assert(Uun = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uuk),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uue),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aa_nat461013784042436401_VEBTi(aTP_Lamm_zb(Uu),Uub),Uuc),Uud),Uuf),Uug),Uui),Uuj),Uul),Uum),Uun)),
        aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uuh),Uui)),Uub,Uuc),Uud)) ) ).

% ATP.lambda_1117
tff(fact_11293_ATP_Olambda__1118,axiom,
    ! [Uu: fun_VE2018681615261037117_VEBTi,Uua: nat,Uub: nat,Uuc: array_VEBT_VEBTi,Uud: vEBT_VEBTi,Uue: nat,Uuf: list_VEBT_VEBT,Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: nat,Uuo: nat] :
      aa_nat5408703056993926195_VEBTi(aa_nat4282640288315822012_VEBTi(aa_nat7653869184832469701_VEBTi(aa_nat2287630409448422734_VEBTi(aa_nat2878941900432676695_VEBTi(aa_nat532351713702512864_VEBTi(aa_nat5473138614813200873_VEBTi(aa_nat354071531552010866_VEBTi(aa_VEB545764574255792529_VEBTi(aa_lis7527251260247064854_VEBTi(aa_nat2109960713227968087_VEBTi(aa_VEB4528852985946928827_VEBTi(aa_arr8719208259821620636_VEBTi(aa_nat4030471164133998385_VEBTi(aa_nat552572580330102142_VEBTi(aTP_Lamm_wl(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun),Uuo) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uun),Uuo)
        & ~ ( ( Uua = Uuh )
            | ( Uua = Uui ) ) ),
        heap_T7982501707604696571_VEBTi(refine_Imp_assert(Uun = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uuk),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uue),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat7312404184930824517_VEBTi(aa_nat3221004963518969490_VEBTi(aa_nat1817389164722854239_VEBTi(aa_nat317164691840817580_VEBTi(aa_nat3080504576880736121_VEBTi(aa_VEB4815025940789304112_VEBTi(aa_lis3577763594685753089_VEBTi(aa_VEB8209137959338826235_VEBTi(aa_arr747291131267414768_VEBTi(aa_nat461013784042436401_VEBTi(aTP_Lamm_wk(Uu),Uub),Uuc),Uud),Uuf),Uug),Uui),Uuj),Uul),Uum),Uun)),
        aa_VEB836174949890266638_VEBTi(heap_T3630416162098727440_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uuh),Uui)),Uub,Uuc),Uud)) ) ).

% ATP.lambda_1118
tff(fact_11294_ATP_Olambda__1119,axiom,
    ! [Uu: option4927543243414619207at_nat,Uua: nat,Uub: list_VEBT_VEBT,Uuc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_lis4184305612057247525VEBT_o(aa_nat8798449810042143974VEBT_o(aa_opt7072554309604550194VEBT_o(aTP_Lamm_iq,Uu),Uua),Uub),Uuc)
    <=> $true ) ).

% ATP.lambda_1119
tff(fact_11295_ATP_Olambda__1120,axiom,
    ! [Uu: vEBT_VEBT,Uua: real] : aa_real_VEBT_VEBT(aTP_Lamm_agb(Uu),Uua) = Uu ).

% ATP.lambda_1120
tff(fact_11296_ATP_Olambda__1121,axiom,
    ! [Uu: vEBT_VEBT,Uua: $o] : aa_o_VEBT_VEBT(aTP_Lamm_agc(Uu),(Uua)) = Uu ).

% ATP.lambda_1121
tff(fact_11297_ATP_Olambda__1122,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat] : aa_nat_VEBT_VEBT(aTP_Lamm_age(Uu),Uua) = Uu ).

% ATP.lambda_1122
tff(fact_11298_ATP_Olambda__1123,axiom,
    ! [Uu: vEBT_VEBT,Uua: int] : aa_int_VEBT_VEBT(aTP_Lamm_agf(Uu),Uua) = Uu ).

% ATP.lambda_1123
tff(fact_11299_ATP_Olambda__1124,axiom,
    ! [Uu: heap_T8145700208782473153_VEBTi,Uua: product_unit] : aa_Pro3581650104493980720_VEBTi(aTP_Lamm_kp(Uu),Uua) = Uu ).

% ATP.lambda_1124
tff(fact_11300_ATP_Olambda__1125,axiom,
    ! [Uu: heap_T8145700208782473153_VEBTi,Uua: produc3960855945107176009Ti_nat] : aa_Pro721367001817565044_VEBTi(aTP_Lamm_rl(Uu),Uua) = Uu ).

% ATP.lambda_1125
tff(fact_11301_ATP_Olambda__1126,axiom,
    ! [Uu: heap_T8145700208782473153_VEBTi,Uua: fun_Pr4404027214751104635_VEBTi] : aa_fun7494984083220284744_VEBTi(aTP_Lamm_rg(Uu),Uua) = Uu ).

% ATP.lambda_1126
tff(fact_11302_ATP_Olambda__1127,axiom,
    ! [Uu: heap_T2636463487746394924on_nat,Uua: product_unit] : aa_Pro451875010097498139on_nat(aTP_Lamm_km(Uu),Uua) = Uu ).

% ATP.lambda_1127
tff(fact_11303_ATP_Olambda__1128,axiom,
    ! [Uu: heap_T2636463487746394924on_nat,Uua: produc3960855945107176009Ti_nat] : aa_Pro2727408364141080927on_nat(aTP_Lamm_rm(Uu),Uua) = Uu ).

% ATP.lambda_1128
tff(fact_11304_ATP_Olambda__1129,axiom,
    ! [Uu: heap_T2636463487746394924on_nat,Uua: fun_Pr342922516890243977on_nat] : aa_fun7698174670542568229on_nat(aTP_Lamm_re(Uu),Uua) = Uu ).

% ATP.lambda_1129
tff(fact_11305_ATP_Olambda__1130,axiom,
    ! [Uu: heap_T2636463487746394924on_nat,Uua: fun_Pr633315466666964198on_nat] : aa_fun1100322450986716680on_nat(aTP_Lamm_rh(Uu),Uua) = Uu ).

% ATP.lambda_1130
tff(fact_11306_ATP_Olambda__1131,axiom,
    ! [Uu: heap_Time_Heap_o,Uua: product_unit] : aa_Pro3379321066291785715Heap_o(aTP_Lamm_ko(Uu),Uua) = Uu ).

% ATP.lambda_1131
tff(fact_11307_ATP_Olambda__1132,axiom,
    ! [Uu: heap_Time_Heap_o,Uua: produc3960855945107176009Ti_nat] : aa_Pro6173831900156065711Heap_o(aTP_Lamm_rn(Uu),Uua) = Uu ).

% ATP.lambda_1132
tff(fact_11308_ATP_Olambda__1133,axiom,
    ! [Uu: heap_Time_Heap_o,Uua: fun_Pr238758252084143515Heap_o] : aa_fun6848502500920544641Heap_o(aTP_Lamm_rf(Uu),Uua) = Uu ).

% ATP.lambda_1133
tff(fact_11309_ATP_Olambda__1134,axiom,
    ! [Uu: heap_Time_Heap_o,Uua: fun_Pr940977554174145406Heap_o] : aa_fun3870618101526761188Heap_o(aTP_Lamm_ri(Uu),Uua) = Uu ).

% ATP.lambda_1134
tff(fact_11310_ATP_Olambda__1135,axiom,
    ! [Uu: heap_Time_Heap_nat,Uua: product_unit] : aa_Pro4051170119821641419ap_nat(aTP_Lamm_kn(Uu),Uua) = Uu ).

% ATP.lambda_1135
tff(fact_11311_ATP_Olambda__1136,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_uz,Uu),Uua) = Uu ).

% ATP.lambda_1136
tff(fact_11312_ATP_Olambda__1137,axiom,
    ! [Uu: assn,Uua: product_unit] : aa_Product_unit_assn(aTP_Lamm_aqn(Uu),Uua) = Uu ).

% ATP.lambda_1137
tff(fact_11313_ATP_Olambda__1138,axiom,
    ! [Uu: assn,Uua: produc6575502325842934193n_assn] : aa_Pro2979818503705714077n_assn(aTP_Lamm_aga(Uu),Uua) = Uu ).

% ATP.lambda_1138
tff(fact_11314_ATP_Olambda__1139,axiom,
    ! [Uu: assn,Uua: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(aTP_Lamm_dr,Uu),Uua) = Uu ).

% ATP.lambda_1139
tff(fact_11315_ATP_Olambda__1140,axiom,
    ! [Uu: real,Uua: vEBT_VEBT] : aa_VEBT_VEBT_real(aTP_Lamm_afz(Uu),Uua) = Uu ).

% ATP.lambda_1140
tff(fact_11316_ATP_Olambda__1141,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_va,Uu),Uua) = Uu ).

% ATP.lambda_1141
tff(fact_11317_ATP_Olambda__1142,axiom,
    ! [Uu: $o,Uua: nat] :
      ( aa_nat_o(aa_o_fun_nat_o(aTP_Lamm_agd,(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_1142
tff(fact_11318_ATP_Olambda__1143,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : aa_VEBT_VEBT_nat(aTP_Lamm_afy(Uu),Uua) = Uu ).

% ATP.lambda_1143
tff(fact_11319_ATP_Olambda__1144,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dq,Uu),Uua) = Uu ).

% ATP.lambda_1144
tff(fact_11320_ATP_Olambda__1145,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ds,Uu),Uua) = Uu ).

% ATP.lambda_1145
tff(fact_11321_ATP_Olambda__1146,axiom,
    ! [Uu: fun_na5556145776640407093on_nat,Uua: produc3625547720036274456_VEBTi] : aa_Pro1290670013494283871on_nat(aTP_Lamm_rj(Uu),Uua) = Uu ).

% ATP.lambda_1146
tff(fact_11322_ATP_Olambda__1147,axiom,
    ! [Uu: fun_na2343073368905879023Heap_o,Uua: produc3625547720036274456_VEBTi] : aa_Pro259339148018954309Heap_o(aTP_Lamm_rk(Uu),Uua) = Uu ).

% ATP.lambda_1147
tff(fact_11323_ATP_Olambda__1148,axiom,
    ! [Uu: fun_na1476630412504255114at_nat,Uua: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_hz(Uu),Uua) = Uu ).

% ATP.lambda_1148
tff(fact_11324_ATP_Olambda__1149,axiom,
    ! [Uu: fun_nat_o,Uua: nat] : aa_nat_fun_nat_o(aTP_Lamm_ib(Uu),Uua) = Uu ).

% ATP.lambda_1149
tff(fact_11325_ATP_Olambda__1150,axiom,
    ! [Uu: fun_in4211310926729683942nt_int,Uua: int] : aa_int2565445551602696444nt_int(aTP_Lamm_id(Uu),Uua) = Uu ).

% ATP.lambda_1150
tff(fact_11326_ATP_Olambda__1151,axiom,
    ! [Uu: fun_int_o,Uua: int] : aa_int_fun_int_o(aTP_Lamm_if(Uu),Uua) = Uu ).

% ATP.lambda_1151
tff(fact_11327_ATP_Olambda__1152,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_fun_int_int(aTP_Lamm_ih(Uu),Uua) = Uu ).

% ATP.lambda_1152
tff(fact_11328_ATP_Olambda__1153,axiom,
    ! [Uu: assn,Uua: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(aTP_Lamm_du,Uu),Uua) = Uua ).

% ATP.lambda_1153
tff(fact_11329_ATP_Olambda__1154,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dt,Uu),Uua) = Uua ).

% ATP.lambda_1154
tff(fact_11330_ATP_Olambda__1155,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_dv,Uu),Uua) = Uua ).

% ATP.lambda_1155
tff(fact_11331_ATP_Olambda__1156,axiom,
    ! [Uu: $o,Uua: $o] : aa_o_P8223745691486049304T_VEBT(aa_o_f7062522455407347339T_VEBT(aTP_Lamm_ba,(Uu)),(Uua)) = undefi7074909574607331924T_VEBT ).

% ATP.lambda_1156
tff(fact_11332_ATP_Olambda__1157,axiom,
    ! [Uu: $o,Uua: $o] :
      ( aa_o_o(aa_o_fun_o_o(aTP_Lamm_ir,(Uu)),(Uua))
    <=> $false ) ).

% ATP.lambda_1157
tff(fact_11333_ATP_Olambda__1158,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_hm,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1158
tff(fact_11334_ATP_Olambda__1159,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_hn,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1159
tff(fact_11335_ATP_Olambda__1160,axiom,
    ! [Uu: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aTP_Lamm_ec,Uu) = Uu ).

% ATP.lambda_1160
tff(fact_11336_ATP_Olambda__1161,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_aod,Uu) = Uu ).

% ATP.lambda_1161
tff(fact_11337_ATP_Olambda__1162,axiom,
    ! [Uu: uint32] : aa_uint32_uint32(aTP_Lamm_ix,Uu) = Uu ).

% ATP.lambda_1162
tff(fact_11338_ATP_Olambda__1163,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_iy,Uu) = Uu ).

% ATP.lambda_1163
tff(fact_11339_ATP_Olambda__1164,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_iz,Uu) = Uu ).

% ATP.lambda_1164
tff(fact_11340_ATP_Olambda__1165,axiom,
    ! [Uu: num] : aa_num_num(aTP_Lamm_ed,Uu) = Uu ).

% ATP.lambda_1165
tff(fact_11341_ATP_Olambda__1166,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_eb,Uu) = Uu ).

% ATP.lambda_1166
tff(fact_11342_ATP_Olambda__1167,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ja,Uu) = Uu ).

% ATP.lambda_1167
tff(fact_11343_ATP_Olambda__1168,axiom,
    ! [Uu: fun_fu6749172196426063865uint32] : aa_fun3431294132209178672uint32(aTP_Lamm_dw,Uu) = Uu ).

% ATP.lambda_1168
tff(fact_11344_ATP_Olambda__1169,axiom,
    ! [Uu: fun_fu3466578569608634614eger_o] : aa_fun1804373999164127658eger_o(aTP_Lamm_dx,Uu) = Uu ).

% ATP.lambda_1169
tff(fact_11345_ATP_Olambda__1170,axiom,
    ! [Uu: fun_fu8139443716560175030uint32] : aa_fun3432716020469233706uint32(aTP_Lamm_dy,Uu) = Uu ).

% ATP.lambda_1170
tff(fact_11346_ATP_Olambda__1171,axiom,
    ! [Uu: fun_fu1303640425391265102uint32] : aa_fun3449071308854940378uint32(aTP_Lamm_dz,Uu) = Uu ).

% ATP.lambda_1171
tff(fact_11347_ATP_Olambda__1172,axiom,
    ! [Uu: fun_fu5030138910357066062uint32] : aa_fun953596214170897114uint32(aTP_Lamm_ea,Uu) = Uu ).

% ATP.lambda_1172
tff(fact_11348_ATP_Olambda__1173,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_agh,Uu) = zero_zero_complex ).

% ATP.lambda_1173
tff(fact_11349_ATP_Olambda__1174,axiom,
    ! [Uu: word_N3645301735248828278l_num1] : aa_wor7274019993978516266l_num1(aTP_Lamm_is,Uu) = zero_z3563351764282998399l_num1 ).

% ATP.lambda_1174
tff(fact_11350_ATP_Olambda__1175,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_it,Uu) = zero_zero_real ).

% ATP.lambda_1175
tff(fact_11351_ATP_Olambda__1176,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_iu,Uu) = zero_zero_rat ).

% ATP.lambda_1176
tff(fact_11352_ATP_Olambda__1177,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_um,Uu) = zero_zero_complex ).

% ATP.lambda_1177
tff(fact_11353_ATP_Olambda__1178,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_tz,Uu) = zero_zero_real ).

% ATP.lambda_1178
tff(fact_11354_ATP_Olambda__1179,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_iv,Uu) = zero_zero_nat ).

% ATP.lambda_1179
tff(fact_11355_ATP_Olambda__1180,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_ua,Uu) = zero_zero_int ).

% ATP.lambda_1180
tff(fact_11356_ATP_Olambda__1181,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_iw,Uu) = zero_zero_int ).

% ATP.lambda_1181
tff(fact_11357_ATP_Olambda__1182,axiom,
    ! [Uu: product_unit] : aa_Product_unit_assn(aTP_Lamm_aro,Uu) = one_one_assn ).

% ATP.lambda_1182
tff(fact_11358_ATP_Olambda__1183,axiom,
    ! [Uu: heap_e7401611519738050253t_unit] : aa_hea6347871425501556311it_nat(aTP_Lamm_sp,Uu) = none_P7832717587476222275it_nat ).

% ATP.lambda_1183
tff(fact_11359_ATP_Olambda__1184,axiom,
    ! [Uu: heap_e7401611519738050253t_unit] : aa_hea5089399211691266754it_nat(aTP_Lamm_rp,Uu) = none_P1551326421579882414it_nat ).

% ATP.lambda_1184
tff(fact_11360_ATP_Olambda__1185,axiom,
    ! [Uu: heap_e7401611519738050253t_unit] : aa_hea8154171661607227764it_nat(aTP_Lamm_so,Uu) = none_P7668321371905463026it_nat ).

% ATP.lambda_1185
tff(fact_11361_ATP_Olambda__1186,axiom,
    ! [Uu: heap_e7401611519738050253t_unit] : aa_hea8112163199330556402it_nat(aTP_Lamm_sq,Uu) = none_P281974696781278558it_nat ).

% ATP.lambda_1186
tff(fact_11362_ATP_Olambda__1187,axiom,
    ! [Uu: produc120671012495760973it_nat] :
      ( aa_Pro2986953523225089223_nat_o(aTP_Lamm_fp,Uu)
    <=> $false ) ).

% ATP.lambda_1187
tff(fact_11363_ATP_Olambda__1188,axiom,
    ! [Uu: produc8047831477865546771it_nat] :
      ( aa_Pro6395689520723908999_nat_o(aTP_Lamm_fq,Uu)
    <=> $false ) ).

% ATP.lambda_1188
tff(fact_11364_ATP_Olambda__1189,axiom,
    ! [Uu: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_ft,Uu)
    <=> $false ) ).

% ATP.lambda_1189
tff(fact_11365_ATP_Olambda__1190,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_jl,Uu)
    <=> $false ) ).

% ATP.lambda_1190
tff(fact_11366_ATP_Olambda__1191,axiom,
    ! [Uu: complex] :
      ( aa_complex_o(aTP_Lamm_jk,Uu)
    <=> $false ) ).

% ATP.lambda_1191
tff(fact_11367_ATP_Olambda__1192,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_jm,Uu)
    <=> $false ) ).

% ATP.lambda_1192
tff(fact_11368_ATP_Olambda__1193,axiom,
    ! [Uu: num] :
      ( aa_num_o(aTP_Lamm_fr,Uu)
    <=> $false ) ).

% ATP.lambda_1193
tff(fact_11369_ATP_Olambda__1194,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_fs,Uu)
    <=> $false ) ).

% ATP.lambda_1194
tff(fact_11370_ATP_Olambda__1195,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_jn,Uu)
    <=> $false ) ).

% ATP.lambda_1195
tff(fact_11371_ATP_Olambda__1196,axiom,
    ! [Uu: produc120671012495760973it_nat] :
      ( aa_Pro2986953523225089223_nat_o(aTP_Lamm_fk,Uu)
    <=> $true ) ).

% ATP.lambda_1196
tff(fact_11372_ATP_Olambda__1197,axiom,
    ! [Uu: produc8047831477865546771it_nat] :
      ( aa_Pro6395689520723908999_nat_o(aTP_Lamm_fl,Uu)
    <=> $true ) ).

% ATP.lambda_1197
tff(fact_11373_ATP_Olambda__1198,axiom,
    ! [Uu: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_fo,Uu)
    <=> $true ) ).

% ATP.lambda_1198
tff(fact_11374_ATP_Olambda__1199,axiom,
    ! [Uu: num] :
      ( aa_num_o(aTP_Lamm_fm,Uu)
    <=> $true ) ).

% ATP.lambda_1199
tff(fact_11375_ATP_Olambda__1200,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_fn,Uu)
    <=> $true ) ).

% ATP.lambda_1200
tff(fact_11376_ATP_Olambda__1201,axiom,
    ! [Uu: real] : aa_real_fun_nat_nat(aTP_Lamm_lw,Uu) = suc ).

% ATP.lambda_1201
tff(fact_11377_ATP_Olambda__1202,axiom,
    ! [Uu: $o] : aa_o_fun_nat_nat(aTP_Lamm_lx,(Uu)) = suc ).

% ATP.lambda_1202
tff(fact_11378_ATP_Olambda__1203,axiom,
    ! [Uu: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ly,Uu) = suc ).

% ATP.lambda_1203
tff(fact_11379_ATP_Olambda__1204,axiom,
    ! [Uu: int] : aa_int_fun_nat_nat(aTP_Lamm_lz,Uu) = suc ).

% ATP.lambda_1204

% Helper facts (13)
tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X9: int,Y: int] :
      ( ( X9 != Y )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X9: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X9: nat,Y: nat] :
      ( ( X9 != Y )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X9: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X9: real,Y: real] :
      ( ( X9 != Y )
      | aa_real_o(aa_real_fun_real_o(fequal_real,X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X9: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(fequal_real,X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_o] :
      ( aa_real_o(P,fChoice_real(P))
      = ( ? [X6: real] : aa_real_o(P,X6) ) ) ).

tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X9: complex,Y: complex] :
      ( ( X9 != Y )
      | aa_complex_o(fequal_complex(X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X9: complex,Y: complex] :
      ( ~ aa_complex_o(fequal_complex(X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X9: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( X9 != Y )
      | aa_VEBT_VEBT_o(fequal_VEBT_VEBT(X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X9: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(fequal_VEBT_VEBT(X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X9: product_prod_int_int,Y: product_prod_int_int] :
      ( ( X9 != Y )
      | aa_Pro4592068669816069595_int_o(fequal6010348034681699726nt_int(X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X9: product_prod_int_int,Y: product_prod_int_int] :
      ( ~ aa_Pro4592068669816069595_int_o(fequal6010348034681699726nt_int(X9),Y)
      | ( X9 = Y ) ) ).

% Conjectures (2)
tff(conj_0,hypothesis,
    ! [T6: vEBT_VEBT,Ti3: vEBT_VEBTi,X3: nat] : refine7594492741263601813on_nat(aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(vEBT_vebt_predi,Ti3),X3),aa_nat947733127081326238on_nat(aa_VEB3566417204837707778on_nat(aa_VEB1030146501360450209on_nat(f,T6),Ti3),X3)) ).

tff(conj_1,conjecture,
    refine7594492741263601813on_nat(aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aTP_Lamm_ai,aTP_Lamm_aj),tia),aa_VEB2711320435582112505on_nat(vEBT_c6250501799366334488on_nat(aTP_Lamm_bc,aTP_Lamm_aj),tia)) ).

%------------------------------------------------------------------------------